University  of  California  •  Berkeley 

THE  THEODORE  P.  HILL  COLLECTION 

of 

EARLY  AMERICAN  MATHEMATICS  BOOKS 


I 


ELEMENTARY 


ALGEBRA: 


EMBRACING 


THE     FIRST     PRINCIPLES 


OF 


THE   SCIENCE 


BY  CHARLES  DAVIES,  LL.D. 

ATTTHO  B    OF 

ARITHMETIC,    ELEMENTARY   GEOMETRY,    ELEMENTS   OF  SURVEYING, 
ELEMENTS  OF  DESCRIPTIVE  AND  ANALYTICAL  GEOMETRY,  ELE- 
MENTS  OF   DIFFERENTIAL  AND   INTEGRAL  CALCULUS, 
AND  A  TREATISE  ON  SHADES,  SHADOWS, 
AND   PERSPECTIVE- 


NEW   YORK: 

PUBLISHED   BY  A.  S.  BARNES  &  CO. 
CINCINNATI  :-H.  W.  DERBY  &  CO 
1850. 


Entered  according  to  the  Act  of  Congress,  in  the  year  oae  thousand 
eight  hundred  and  Forty-Five,  by  CHARLES  DAVIES,  in  the  Clerk's 
Office  of  the  District  Court  of  the  United  States,  for  the  Southern 
District  of  New  York 


F.  C.  GUTIERREZ, 

PRINTER, 

Cor.  John  and  Dutch-streets,  N.  Y. 


PIIEFACE. 

ALTHOUGH  Algebra  naturally  follows  Arithmetic  in  a  course 
ef  scientific  studies,  yet  the  change  from  numbers  to  a  sys- 
tem of  reasoning  entirely  conducted  by  letters  and  signs  is 
rather  abrupt  and  not  unfrequently  discourages  the  pupil. 

In  this  work  it  has  been  the  intention  to  form  a  connect- 
ing link  between  Arithmetic  and  Algebra,  to  unite  and  blend, 
as  far  as  possible,  the  reasoning  on  numbers  with  the  more 
abstruse  method  of  analysis. 

The  Algebra  of  M.  Bourdon  has  been  closely  followed. 
Indeed,  it  has  been  a  part  of  the  plan,  to  furnish  an  introduc- 
tion to  that  admirable  treatise,  which  is  justly  considered, 
both  in  Europe  and  this  country,  as  the  best  work  on  the 
subject  of  which  it  treats,  that  has  yet  appeared. 

This  work,  however,  even  in  its  abridged  form,  is  too 
voluminous  for  schools,  and  the  reasoning  is  too  elaborate 
and  metaphysical  for  beginners. 

It  has  been  thought  that  a  work  which  should  so  far  mo- 
dify the  system  of  M.  Bourdon  as  to  bring  it  within  the 
scope  of  our  common  schools,  by  giving  to  it  a  more  prac- 
tical and  tangible  form,  could  not  fail  to  be  useful.  Such  is 
the  object  of  the  ELEMENTARY  ALGEBRA. 

Having  within  the  past  year  carefully  revised  the  Algebra 
of  M.  Bourdon,  and  made  therein  many  important  changes 
and  alterations,  both  by  the  addition  of  new  rules  and  in  the 
abridgment  and  simplification  of  those  before  given,  it  be- 
came necessary  to  make  corresponding  changes  in  the  in- 
troductory work.  The  alterations  before  made,  in  the  form 
of  an  Introduction,  the  Treatise  on  Logarithms,  and  the 
Supplement  containing  practical  examples,  with  solutions 
given  in  the  Key,  have  all  been  retained ;  and  the  work  is 
now  presented  to  the  public  in  a  form  which  it  is  hoped  will 
not  require  alteration. 

WBST  POINT,  January,  1845.  ( iii  ) 


DAVIES' 

COURSE  OF  MATHEMATICS. 


DAVIES' TABLE  BOOK. 

DAVIES'  FIRST  LESSONS  IN  ARITHMETIC— 

DAVIES'  ARITHMETIC.— Designed  for  the  use  of  Academies  and 
Schools. 

DAVIES'  GRAMMAR  OF  ARITMETIC. 
KEY  TO  DAVIES'  ARITHMETIC. 

DAVIES'  UNIVERSITY  ARITHMETIC— Embracing  the  Science 
of  Numbers,  and  their  numerous  applications. 

KEY  TO  DAVIES'  UNIVERSITY  ARITHMETIC. 

DAVIES'  ELEMENTARY  ALGEBRA— Being  an  Introduction  to 
the  Science,  and  forming  a  connecting  link  between  ARITHMETIC  and 
ALGEBRA. 

KEY  TO  DAVIES'  ELEMENTARY  ALGEBRA. 

DAVIES'  ELEMENTARY  GEOMETRY.— This  work  embraces  the 
elementary  principles  of  Geometry.  The  reasoning  is  plain  and  con- 
cise, but  at  the  same  time  strictly  rigorous. 

DAVIES'  ELEMENTS   OF  DRAWING  AND  MENSURATION 

— Applied  to  the  Mechanic  Arts. 

DAVIES'  BOURDON'S  ALGEBRA— Including  Sturms'  Theorem,— 
Being  an  Abridgment  of  the  work  of  M.  Bourdon,  with  the  addition  of 
practical  examples 

DAVIES'   LEGENDRE'S   GEOMETRY  AND  TRIGONOMETRY. 

— Being  an  Abridgment  of  the  work  of  M.  Legendre,  with  the  addition 
of  a  Treatise  on  MENSURATION  OF  PLANES  AND  SOLIDS,  and  a  Table  of 
LOGARITHMS  and  LOGARITHMIC  SINES. 

DAVIES'  SURVEYING— With  a  description  and  plates  of  the  THEOD- 
OLITE,  COMPASS,  PLANE-TABLE,  and  LEVEL  :  also,  Maps  of  the  TOPO- 
GRAPHICAL SIGNS  adopted  by  the  Engineer  Department — an  explana- 
tion of  the  method  of  surveying  the  Public  Lands,  and  an  Elementary 
Treatise  on  NAVIGATION. 

DAVIES'   ANALYTICAL   GEOMETRY— Embracing    the    EQUA 

TIONS    OF    THE    PoiNT    AND    STRAIGHT  LlNE of  the  CoNIC  SECTIONS of 

the  LINE  AND  PLANE  IN  SPACE — also,  the  discussion  of  the  GENERAL 
EQUATION  of  the  second  degree,  and  of  SURFACES  of  the  second  order. 

DAVIES'  DESCRIPTIVE  GEOMETRY,— With  its  application  U» 
SPHERICAL  PROJECTIONS. 

DAVIES'  SHADOWS  AND  LINEAR  PERSPECTIVE. 
DAVIES'  DIFFERENTIAL  AND  INTEGRAL  CALCULUS 


CONTENTS. 


CHAPTER  I. 

PRELIMINARY    DEFINITIONS    AND    REMARKS. 

ARTICLES. 

Algebra — Definitions — Explanation  of  the  Algebraic  Signs,      -  1 — 23 

Similar  Terms — Reduction  of  Similar  Terms,         -  23 — 26 

Addition— Rule, 26 — 28 

Subtraction — Rule — Remark,        -         -        -        -        -         -  28 — 33 

Multiplication — Rule  for  Monomials,      •  33 — 36 

Rule  for  Polynomials  and  Signs, 36 — 38 

Remarks — Properties  Proved, 38—42 

Division  of  Monomials — Rule,       ------  42 — 45 

Signification  of  the  Symbol  ao,                -         -         -         -        -  45 — 46 

Of  the  Signs  in  Divison,        -         -         -        -         -         -    .     -  46 — 47 

Division  of  Polynomials,        ---.--  47 — 49 

CHAPTER  II. 

ALGEBRAIC    FRACTIONS. 

Definitions — Entire  Quantity — Mixed  Quantity,       -         -  49 — 52 

To  Reduce  a  Fraction  to  its  Simplest  Terms           -  52 

To  Reduce  a  Mixed  Quantity  to  a  Fraction,            ...  63 

To  Reduce  a  Fraction  to  an  Entire  or  Mixed  Quantity,  -  64 

To  Reduce  Fractions  to  a  Common  Denominator,  -  55 

To  Add  Fractions, '     -  56 


VI  CONTENTS. 

ARTICLES. 

To  Subtract  Fractions, 57 

To  Multiply  Fractions, 58 

To  Divide  Fractions, --  69 


CHAPTER  III 

EQUATIONS    OF    THE    FIRST    DEGREE. 

Definition  of  an  Equation — Properties  of  Equations,  -  60 — 66 

Transformation  of  Equations — First  and  Second,  -  66 — 70 

Resolution  of  Equations  of  the  First  Degree — Rule,  -  70 

Questions  involving  Equations  of  the  First  Degree,  -  '-  71 — 72 
Equations  of  the  First  Degree  involving  Two  Unknown 

Quantities,  - -  72 

Elimination — By  Addition — By  Subtraction — Bv  Comparison,  -  73 — 76 
Resolution  of  Questions  involving  Two  or  more  Unknown 

Quantities, 76 — 79 

CHAPTER  IV. 

OF    POWERS. 

Definition  of  Po  .vers,    ----..-.  79 

To  raise  Monomials  to  any  Power,         -  80 

To  raise  Polynomials  to  any  Power,       -  81 

To  raise  a  Fraction  to  any  Power,          -         ....  82 — 83 

Binomial  Theorem,               84—90 

CHAPTER  V. 

Definition    of    Squares — Of    Square    Roots — And    Perfect 

Squares, 90 — 96 

Rule  for  Extracting  the  Square  Root  of  Numbers,        -         -  96 — 100 

Square  Roots  of  Fractions, -  100 — 103 

Squan  s  Roots  of  Monomials, 103 — 107 

Calculus  of  Radicals  of  the  Second  Degree,         -         -         -  107—109 


CONTENTS. 


Vll 


Addition  of  Radicals,         .... 

Subtraction  of  Radicals,    - 

Multiplication  of  Radicals, 

Division  of  Radicals,         .... 

Extraction  of  the  Square  Root  of  Polynomials, 


ARTICLES. 

109 
110 
111 
112 
113—116 


CHAPTER  VI. 

Equations  of  the  Second  Degree,        ... 
Definition  and  Form  of  Equations,      - 
Incomplete  Equations,       - 
Complete  Equations,         ..... 

Four  Forms,    ------- 

Resolution  of  Equations  of  the  Second  Degree,  - 
Properties  of  the  Roots,    -         -        - 


116 

116—118 
118—122 

122 

123—127 
127—128 
128—134 


CHAPTER  VII. 

Of  Progressions,       ........  135 

Progressions  by  Differences,      ......  136 — 138 

Last  Term, 138—140 

Sum  of  the  Extremes — Sum  of  the  Series,          ...  140 — 141 

The  Five  Numbers — To  Find  any  Number  of  Means,          -  141 — 144 

Geometrical  Proportion  and  Progression,    -  144 

Various  Kinds  of  Proportion,     -         -         -                  -  144 — 166 

Geometrical  Progression,  -------  166 

Last  Term — Sum  of  the  Series, 167 — 171 

Progressions  having  an  Infinite  Number  of  Terms,       -         -  171 — 172 

The  Five  Numbers— To  Find  One  Mean   -  172—173 

CHAPTER  VIII. 

Theory  of  Logarithms 174 — 179 


INTRODUCTION. 


LESSON  I. 

1.  JOHN  and  Charles  have  twelve  apples  between  them, 
and  each  has  as  many  as  the  other :  How  many  has  each  ? 

If  we  suppose  the  apples  divided  into  two  equal  parts,  it  is 
plain  that  John  will  have  one  part  and  Charles  the  other : 
hence,  they  will  each  have  six  apples. 

In  Algebra,  we  often  represent  numbers  by  the  letters  of 
the  alphabet ;  that  is,  we  take  a  letter  to  stand  for  a  number. 
Thus,  let  x  stand  for  the  apples  which  John  has.  Then,  as 
Charles  has  an  equal  number,  x  will  also  stand  for  the  apples 
which  he  has.  But  together,  they  have  twelve  apples ;  hence, 
twice  x  must  be  equal  to  12.  This  we  write  thus : 

and  if  twice  x  is  equal  to  12,  it  follows  that  once  a?,  or  a?,  will 
be  equal  to  6.  This  we  write  thus : 

a?=—  =6 

~~2  ~~ 

When  we  write  x  by  itself,  we  mean  one  #,  or  the  same  as 
la?.  If  we  write  2a;,  we  mean  that  x  is  twice  taken ;  if  3a?, 
that  it  is  taken  three  times,  &,c. 

QUEST. — 1.  In  the  first  question,  how  many  apples  has  each  boy? 
By  what  are  numbers  represented  in  Algebra  ?  If  a:  stands  by  itself,  how 
many  times  x  are  expressed?  What  does  Zx  denote?  What  3x1 
What4#,  &c.  If  \vc  have  x-\-x,  to  how  many  times  x  is  it  equal  ?  ff 
we  have  the  value  of  2or,  how  do  we  iiud  the  value  of  x  I 

w 


ELEMENTARY  ALGEBRA 


3.  James  and  John  together  have  24  penches,  and  one  has 
as  many  as  the  other  :  How  many  has  each  ? 

Let  x  stand  for  the  number  of  peaches  which  James  has  : 
then  x  will  also  be  equal  to  the  number  of  peaches  which 
John  has  ;  and  since  they  have  24  between  them, 


that  is,  2a?=24    and    a?==12. 

Therefore,  each  has  twelve  peaches. 

3.  William  and  John  have  36  pears,  and  one  has  as  many 
as  the  other  :  How  many  has  each  ? 

Let  the  number  which  each  has  be  denoted  by  x. 
then  a;+;r=36  ; 

that  is,  2*=36  and  *=-??==*  18. 

4:.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  20  ? 

Let  the  number  be  denoted  by  x  :  then,  as  the  number  is 
to  be  added  to  itself,  we  have 


that  is,  2z=20   or    x=^U=10. 

Hence,  10  is  the  number. 

5.  What  number  is  that  which  added  to  itself  will  give  9 
sum  equal  to  30  ? 

QUEST.  —  2.  In  the  second  question,  what  does  x  stand  for  1  What 
is  twice  x  equal  to  ?  How  then  do  you  find  the  value  of  x  1  3.  In  the 
third  question,  what  does  x  stand  for  1  What  is  x  equal  to  7  How  do 
you  find  the  value  of  x  1  4.  In  the  fourth  question,  what  does  x  stand 
for  7  What  is  twice  x  equal  to  1  How  do  you  then  find  x  1  5.  In  the 
fifth  question,  what  does  x  stand  for  1  How  do  you  find  its  value  1 


INTRODUCTION.  3 

6    What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  60  ? 

7.  What  number  is  that  which  added  to  itself  will  give  a 
sum  equal  to  100  ? 

8.  What  number  added  to  itself  will  give  a  sum  equal  to  80. 

9.  What  number  added  to  itself  will  give  a  sum  equal  to  25. 
10..  What  number  added  to  itself  will  give  a  sum  equal 

to  37J. 


LESSON    II. 

1.  JOHN  and  Charles  together  have  12  apples,  and  Charles 
has  twice  as  many  as  John  :  How  many  has  each  ? 

If  we  now  suppose  the  apples  to  be  divided  into  three  equal 
parts,  it  is  evident  that  John  will  have  one  of  the  parts  and 
Charles  two. 

Let  us  denote  by  x  the  apples  which  John  has.  Then,  2x 
will  be  equal  to  what  Charles  has,  and  x+2x  will  be  equal 
to  all  the  apples.  This  equality  is  thus  expressed  : 


that  is,  3#=>12    or    ;r==4 

o 

therefore,  John  has  4  apples,  and  Charles  8. 


QUEST.  —  6.  How  do  you  find  the  value  of  x  in  the  6th  question  t 
8.  How  in  the  8th  1  9.  How  in  the  9th  ?  1O.  How  in  the  10th  1 

QUESTIONS  ox  LESSON  IT.  —  1.  Into  how  many  parts  may  we  suppose 
the  12  apples  to  be  divided  1  How  many  of  the  parts  will  John  have! 
What  is  the  value  of  each  part  1  If  a?  stands  for  one  of  the  parts,  what 
will  stand  for  two  parts'?  What  for  three  parts'?  If  you  have  the  value 
of  3ar,  how  will  you  find  the  value  of  x  1 


4  ELEMENTARY  ALGEBRA 

2.  James  and  John  have  30  pears,  and  John  has  twice  as 
many  as  James  :  How  many  has  each  ? 

Here,  again,  let  us  suppose  the  whole  number  to  be  divided 
into  three  equal  parts,  of  which  James  must  have  one  part, 
and  John  two. 

Let  iis  then  denote  by  #,  the  number  of  pears  which  James 
has:  then  2x  will  be  equal  to  the  number  of  pears,  which  * 
John  has,  and  x-\-2x  will  be  equal  to  the  whole  number  of 
pears  :  and  we  shall  have 


on 

that  is,  3a?—  30    or    #=—  =  10. 

3 

3.  William  and  John  have  48  quills  between  them,  and 
John  has  twice  as  many  as  William  :  How  many  has  each  ? 

Let  the  number  of  quills  which  William  has  be  denoted  by 
./"  :  then,  since  John  has  twice  as  many,  his  will  be  denoted 
by  2#,  and  the  quills  possessed  by  both  of  them,  by  x-{-2xf 
Hence,  we  shall  have 


t  hat  is,  3^=48    or    #==16. 

O 

[Tence,  William  has  16  quills,  and  John  32. 

4.  What  number  is  that  which  added  to  twice  itself,  will 
jive  a  sum  equal  to  60  ? 

Let  the  number  sought  be  denoted  by  #,  then  twice  the 
number  will  be  denoted  by  2#,  and  we  shall  have 


fin 

that  is,  3#=60    or    a=_=20; 

o 

and  we  see  that  20  added  to  twice  itself  will  give  60. 


QUEST. — 2.  In  question  second,  what  is  the  value  of  one  of  the  parts? 
3-  What  in  question  3rd  1     4.  How  do  you  state  question  4th  1 


INTRODUCTION.  5 

5.  John  says  to  Charles,  "  give  me  your  marbles  and  I  shall 
nave  three  times  as  many  as  I  have  now."  "  No,"  says 
Charles,  «  but  give  me  yours,  and  I  shall  have  just  51."  How 
many  had  each  ? 

Let  the  number  of  marbles  which  John  has  be  denoted  by 
x  :  then,  2x  will  denote  the  number  which  Charles  has,  and 
since  they  have  51  in  all,  we  write 


that  is,  3x=5l    or    #=-—=17. 

o 

6.  What  number  is  that  which  added  to  twice  itself  will 
give  a  sum  equal  to  75  ? 

Let  the  number  be  denoted  by  x  :  then,  twice  the  number 
will  be  expressed  by  2#,  and 


that  is,  3#=75    and    ar==25. 

o 

7.  What   number   added  to  twice  itself  will  give  a  sum 
equal  to  90  ? 

8.  What   number   added  to  twice  itself  will  give  a  sum 
equal  to  57  ? 

9.  What  number   added  to  twice  itself  will  give  a  sum 
equal  to  39  ? 

10.  What  number  added  to  twice  itself  will  give  a  sum 
equal  to  21  ?  _ 

LESSON   III. 

1.  If  James  and  John  together  have  24  quills,  and  John 
has  three  times  as  many  as  James,  how  many  will  each  have  ? 

QUEST.  —  5-  How  do  you  state  question  5th  1  '  6-  Explain  the  6th  ques- 
tion! %  Also  the  7th.  8.  What  is  the  required  number  in  the  8th  1 
9.  What  in  the  9th  f  1O.  What  in  the  10th  1 

2 


5  ELEMENTARY  ALGEBRA. 

It  is  plain  that  if  we  suppose  the  twenty-four  quills  to  be 
divided  in  four  equal  parts,  that  James  will  have  one  of  the 
parts,  and  John  three. 

Let  us  now  designate  by  x  the  number  of  quills  which 
James  has  :  then  3x  will  denote  the  number  of  quills  which 
John  has,  and  we  shall  have 


24 
that  is,  4#i=24    and    a?=  —  =6. 

2.  What  number  is  that  which  added  to  three  times  itself 
will  give  a  sum  equal  to  48  ? 

If  we  denote  the  number  by  a?,  we  shall  have 


that  is,  4#=48   and    #=—  =  12. 

3.  John  and  Charles  have  60  apples  between  them,  and 
Charles  has  three  times  as  many  as  John  :  How  many  has 
each  ? 

If  we  suppose  the  number  of  apples  to  be  divided  into  four 
equal  parts,  it  is  evident  that  John  will  have  one  of  those 
parts,  and  Charles  three. 

Let  x=  the  apples  which  John  has  :  then  3x  will  stand  for 
the  apples  which  Charles  has,  and  we  shall  have 


60 
that  is,  4#=60  and  #=—=15. 

Hence,  John  will  have  15  and  Charles  45. 

QUEST.  —  1.  If  the  24  quills  be  divided  into  four  equal  parts,  how 
many  parts  will  John  have  1  How  many  will  James  have  ]  What  is 
each  part  equal  to  1  £.  If  three  times  a  number  be  added  to  the  number, 
how  many  times  will  the  number  be  taken  1  If  4.r  is  equal  to  48,  what 
is  the  value  of  x  1  3.  Explain  the  third  question  !  If  4.r  is  equal  to 
60,  how  do  you  find  the  value  of  a:  1 


INTRODUCTION. 


4.  What  number  is  that  which  being  added  to  three  times 
itself  will  give  a  sum  equal  to  100  ? 
Let  the  number  be  denoted  by  x  :  then 


inn 

that  is,  4c=100    and    a?=±_=25. 

4 

5.  What  number  is  that  which  if  added  to  four  times  itself, 
the  sum  will  be  equal  to  60  ? 

Let  a?  denote  the  number.     Then, 


that  is,  5z=60    and    *==12. 

0 

6.  What  number  is  that  which  being  multiplied  by  3,  and  the 
product  added  to  twice  the  number  will  give  a  sum  equal  to  75  ? 

Let  the  number  be  denoted  by  x. 
Then,     3x=  the  product  of  the  number  by  3  ; 
and         2x—  twice  the  number; 
and  3x+2x=5x=75-y 

and  a?=  —  =  15,  the  required  number. 

5 

7.  What  number  is  that  which  being  added  to  three  times 
itself  will  give  a  sum  equal  to  140  ? 

8.  What  number  is  that  which  being  multiplied  by  5,  and 
the  product  added  to  the  number,  will  give  a  sum  equal  to  240  ? 

9.  What  number  is  that  which  being  multiplied  by  2,  and 
then  by  3,  and  the  products  added  together,  will  give  a  sum 
equal  to  125  ? 

QUEST.  —  5.  If  a  number  be  added  to  four  times  itself,  how  many  times 
will  the  number  be  taken  !  6.  If  x  stands  for  any  number,  what  will 
stand  for  three  times  that  number  1  What  for  twice  the  number]  T. 
Explain  the  7th  question  1  How  do  you  state  it  1  What  is  4x  equal  to  1 
Why  1  How  then  do  you  find  x  1  8.  How  do  you  state  the  8th  question  ? 
What  is  Qx  equal  to  1  How  then  do  you  find  xl  9.  If  x  denotes  a  number, 
what  will  stand  for  twice  the  number  1  What  for  three  times  the  number  ? 


ELEMENTARY  ALGEBRA. 


LESSON     IV. 

1.  John  and  Charles  together  have  80  apples,  and  Charles 
has  four  times  as  many  as  John  :  How  many  has  each  ? 

If  we  suppose  the  80  apples  to  be  divided  into  5  equal 
parts,  it  is  evident  that  John  will  have  one  of  the  parts,  and 
Charles  four. 

Let  x  stand  for  the  apples  which  John  has  :  then  4#  wilJ 
stand  for  the  apples  which  Charles  has  ;  and 


on 

that  is,  5x=SO    and    x=—=16. 

5 

2.  What  number  added  to  four  times  itself  will  give  a  sum 
equal  to  90  ? 

3.  What  number  added  to  five  times  itself  will  give  a  sum 
equal  to  120  ? 

4.  What  number  added  to  six  times  itself  will  give  a  sum 
equal  to  245  ? 

5.  What  number  added  to  seven  times  itself  will  give  a 
sum  equal  to  360  ? 

N.  B.  The  questions  in  the  preceding  Lessons  will  give 
some  idea  of  the  questions  to  which  Algebra  may  be  applied. 


QUEST. — 1.  If  x  stands  for  John's  apples,  what  will  denote  Charles'  1 
What  will  stand  for  the  apples  which  they  both  have  1  If  nx  is  equal  to 
80,  what  will  x  be  equal  to  1  2.  If  a  number  be  added  to  4  times  itself,  how 
many  times  will  the  number  be  taken  1  If  5  times  a  number  is  equal  to 
90,  what  is  the  value  of  the  number]  3-  Explain  example  3rd.  4. 
Explain  question  4th  1  What  does  x  stand  for  ?  5.  Explain  the  5th 
question  ? 


ELEMENTARY 

ALGEBRA. 

CHAPTER  I. 
Preliminary  Definitions  and  Remarks.      * 

1.  QUANTITY  is  a  general  term  applied  to  every  thing 
which  can  be  estimated  or  measured. 

2.  MATHEMATICS  is  the  science  of  quantity. 

3.  ALGEBRA  is  that  branch  of  mathematics  in  which  the 
quantities  considered  are  represented  by  letters,  and  the  ope- 
rations to  be  performed  upon  them  are  indicated  by  signs. 
These  letters  and  signs  are. called  symbols. 

4.  The  sign  +  ,  is  called  plus ;  and  indicates  the  addition 
of  two  or  more  quantities.     Thus,  9+5,  is  read,  9  plus  5, 
or  9  augmented  by  5. 

If  we  represent  the  number  nine,  by  the  letter  a,  and 
the  number  5  by  the  letter  &,  we  shall  have  a+b,  which  is 
read,  a  plus  b  ;  and  denotes  that  the  number  represented  by 
a  is  to  be  added  to  the  number  represented  by  b. 

5.  The  sign — ,  is  called  minus ;  and  indicates  that  one 


QUEST. — 1,.  What  is  quantity  1    2.  What  is  Mathematics  1     3.  What 
is  Algebra  1     What  are  these  letters  and  signs  called  1     4.  What  does  the 
sign  plus  indicate 1     5.  What  does  the  sign  minus  indicate  7 
2* 


10  ELEMENTARY  ALGEBRA. 

quantity  is  to  be  subtracted  from  another.     Thus,  9  —  5  is 
read,  9  minus  5,  or  9  diminished  by  5. 

In  like  manner,  a  —  6,  is  read,  a  minus  6,  or  a  diminished 
by  b. 

6.  The  sign  x  ,  is  called  the  sign  of  multiplication  ;  and 
when  placed  between  two  quantities,  it  denotes  that  they 
are  to  be  multiplied  together.     The  multiplication  of  two 
quantities  is  also  frequently  indicated  by  simply  placing  a 
point  between  them.     Thus,  36  x  25,  or  36.25,  is  read,  36 
multiplied  by  25,  or  the  product  36  by  25. 

7.  The  multiplication  of  quantities,  which  are  represented 
by  letters,  is  indicated  by  simply  writing  them  one  after  the 
other,  without  interposing  any  sign. 

Thus,  ab  signifies  the  same  thing  as  a  x  b,  or  as  a.b  ; 
and  abc  the  same  as  a  x  b  x  c,  or  as  a.b.c.  Thus,  if  we 
suppose  o  =  36,  and  6=25,  we  have 


Again,  if  we  suppose  a  =2,  6=3  and  c=4,  we  have 
abc=2x  3x4=24. 

It  is  most  convenient  to  arrange  the  letters  of  a  product 
in  alphabetical  order. 

8.  In  the  product  of  several  letters,  as  abc,  the  single  let- 
ters, a,  b,  and  c,  are  called  factors  of  the  product.  Thus, 
in  the  product  06,  there  are  two  factors,  a  and  b  ;  in  the 
product  obey  there  are  three,  a,  6,  and  c. 


QUEST. — 6.  What  is  the  sign  of  multiplication  1  "What  does  the  sign 
of  multiplication  indicate  1  In  how  many  ways  may  multiplication  be 
expressed  1  7.  If  letters  only  are  used,  how  may  their  multiplication  be 
expressed?  8.  In  the  product  of  several  letters,  what  is  each  letter 
called  1  How  many  farror*  in  uh  ?--Tn  abr.  ? — In  ahrd  J — In  abcdfJ 


DEFINITION    OF    TERMS.  11 

9.  There  are  three  signs  used  to  denote  division.     Thus 
a-r-  b  denotes  that  a  is  to  be  divided  by  b. 
j-        denotes  that  a  is  to  be  divided  by  b. 
a\b      denotes  that  a  is  to  be  divided  by  b. 

1  0.  The  sign  —  ,  is  called  the  sign  of  equality,  and  is 
read,  is  equal  to.  When  placed  between  two  quantities,  it 
denotes  that  they  are  equal  to  each  other.  Thus,  9  —  5=4  : 
that  is,  9  minus  5  is  equal  to  4  :  Also,  a-\-b—c,  denotes  that 
the  sum  of  the  quantities  a  and  b  is  equal  to  c. 

If  we  suppose  a  =  10,  and  b  =  5,  we  have 

a+b  =  c,     and     10  +  5=c  =  15. 


1  1.  The  sign  >,  is  called  the  sign  of  inequality,  and  is 
used  to  express  that  one  quantity  is  greater  or  less  than 
another. 

Thus,  a  >  b  is  read,  a  greater  than  b  ;  and  a  <  b  is 
read,  a  less  than  b  ;  that  is,  the  opening  of  the  sign  is  turned 
towards  the  greater  quantity.  Thus,  if  a  =9,  and  b=4,  we 
write,  9>4. 

12.  If  a  quantity  is  added  to  itself  several  times  as 
c+a-f  a+a-fa+a,  we  generally  write  it  but  once,  and 
then  place  a  number  before  it  to  show  how  many  times  it 
is  taken.  Thus, 

51      fc  4  H 


QUEST. — 9.  How  many  signs  are  used  in  division "?  What  are  they  ? 
10.  What  is  the  sign  equality  1  When  placed  between  two  quantities, 
what  does  it  indicate1  11.  For  what  is  the  sign  of  inequality  usedl 
Which  quantity  is  placed  on  the  side  of  the  opening  1  12.  What  is  a  co- 
efficient 1  How  many  times  is  ab  taken  in  the  expression  ab  1  In  Bab  1 
In  4ab  1  In  bob  1  In  Gab  1  If  no  co-efficient  is  written,  what  co  efficient 
is  undorstood ' 


12  ELEMENTARY  ALGEBRA. 

The  number  5  is  called  the  co-efficient  of  a,  and  denotes 
that  a  is  taken  5  times. 

When  the  co-efficient  is  1  it  is  generally  omitted.  Thus, 
a  and  la  are  the  same,  each  being  equal  to  a,  or  to  one  a. 

13.  If  a  quantity  be  multiplied  continually  by  itself,  as 
axaXaXaXa,  we  generally  express  the  product  by  writing 
the  letter  once,  and  placing  a  number  to  the  right  of,  and  a 
little  above  it  :  thus, 

aXaXaXaXa=a5. 

The  number  5  is  called  the  exponent  of  a,  and  denotes 
the  number  of  times  which  a  enters  into  the  product  as  a 
factor.  For  example,  if  we  have  a3,  and  suppose  a  =3, 
we  write, 


If  a=4,  «3=43 

and  for  a=5,       a3=53  =  5x5  X  5  =  125. 

If  the  exponent  is  1  it  is  generally  omitted.     Thus,  a1  is 
the  same  as  a,  each  expressing  a  to  the  first  power. 

1  4.  The  power  of  a  quantity  is  the  product  which  results 
from  multiplying  the  quantity  by  itself.    Thus,  in  the  example 
a3_43_4X4X4_64j 

64  is  the  third  power  of  4,  and  the  exponent  3  shows  the 
degree  of  the  power. 

15.  The  sign  •>/     ,  is  called  the  radical  sign,  and  when 


QUEST. — 13.  What  does  the  exponent  of  a  letter  show?  How  many 
times  is  a  a  factor  in  02  ]  In  «s  ]  In  a4 1  In  afi  ?  If  no  exponent  is 
written,  what  exponent  is  understood  1  14.  What  is  the  power  of  a 
quantity  1  What  is  the  third  power  of  2 1  Express  the  4th  power  of  a. 
15.  Express  the  square  root  of  a  quantity.  Also  the  cube  root.  Also 
the  4th  root. 


DEFINITION    OF    TERMS.  13 

prefixed  to  a  quantity,  indicates  that  its  root  is  to  be  ex- 
tracted     Thus, 

•y/o~or  simply  -/^"denotes  the  square  root  of  a. 
^/a  denotes  the  cube  root  of  a. 
-y/a"  denotes  the  fourth  root  of  a. 

The  number  placed  over  the  radical  sign  is  called  the  in- 
dex of  the  root.  Thus,  2  is  the  index  of  the  square  root,  3 
of  the  cube  root,  4  of  the  fourth  root,  &c. 

If  we  suppose    a  =  64,    we  have 


16.  Every  quantity  written  in  algebraic  language,  that 
is,  with  the  aid  of  letters  and  signs,  is  called  an  algebraic 
quantity,  or  the  algebraic  expression  of  a  quantity.  Thus, 

o    f  is  the  algebraic  expression  of  three  times 

I      the  number  a  ; 

2  (  is  the  algebraic  expression  of  five  times 
<      the  square  of  a  ; 

^  is  the  algebraic  expression  of  seven  times 
7a3b2  <      the  product  of  the  cube  of  a  by  the  square 

*       of  6; 

„   _   ,  (  is  the  algebraic  expression  of  the  difference 
\      between  three  times  a  and  five  times  b  ; 

{is  the  algebraic  expression  of  twice  the 
square  of  a,  diminished  by  three  times 
the  product  of  a  by  b,  augmented  by  four 
times  the  square  of  b. 

1.  Write  three  times  the  square  of  a  multiplied  by  the 
cube  of  b.  Ans. 


QUEST. — 16.  What  is  an  algebraic  quantity  1     Is  5ab  an 
quantity  1     Is  9a  !     Is  4y  1     Is  3b  —  xl 


ELEMENTARY  ALGEBRA. 

2.  Write  nine  times  the  cube  of  a  multiplied  by  b,  dimin- 
ished by  the  square  of  c  multiplied  by  d.      Ans.  9a3b  —  c2d. 

3.  If  a=2,  b  =  3,  and  c  =  5,  what  will  be  the  value  of 
3a2  multiplied  by  b2  diminished  by  a  multiplied  by  b  multi- 
Dlied  by  c.     We  have 

3a2b2—abc  =  3  X  22  X  32— 2  X  3  X  5  =  78. 

4.  If   a  =  4,   b  =  6,  c=7,   d=8,    what   is  the  value   of 
9a2+fo— ad\  Ans.   154. 

5.  If   a  =  7,   b  =  3,  c  =  7,   d=l,    what   is  the   value   of 
6ad+3b2c—4d2i  Ans.  227 

6.  If  a=5,   b  =  6,  c=6,   d=5,    what   is  the  value   of 
9abc  —  8<zd-|-46c?  Ans.   1564. 

7.  Write  ten  times  the  square  of  a  into  the  cube  of  b  into 
c  square  into  d3. 

1 7 .  When  an  algebraic  quantity  is  not  connected  with 
any  other,  by  the  sign  of  addition  or  subtraction,  it  is  called 
a  monomial,  or  a  quantity  composed  of  a  single  term,  or  sim- 
ply, a  term.     Thus, 

3a,     5a2,     7a3b2, 

are  monomials,  or  single  terms. 

18.  An  algebraic  expression  composed  of  two  or  more 
parts,  separated  by  the  sign  +  or  — ,  is  called  a  polynomial, 
or  quantity  involving  two  or  more  terms.     For  example, 

3a— 5b     and     2a2  —  3cb  +  4b2 
are  polynomials. 

19.  A  polynomial  composed  of  two  terms,  is  called  a  bi- 
nomial ;  and  a  polynomial  of  three  terms  is  called  a  trinomial. 

QUEST. — 17.  What  is  a  monomial"?  Is  3ai  a  monomial  1  18.  What 
is  a  polynomial?  Is  3a — b  a  polynomial!  19.  What  is  a  binomial! 
What  is  a  trinomial  1 


DEFINITION    OF    TERMS.  15 

20.  Eacl  of  the  literal  factors  which  compose  a  term  is 
called  a  dimension  of  this  term :  and  the  degree  of  a  term  is 
the  number  of  these  factors  or  dimensions.     Thus, 

S  is  a  term  of  one  dimension,  or  of  the  first 
3a  < 

f      degree. 

_  ,  (  is  a  term  of  two  dimensions,  or  of  the 
(      second  degree. 

is    of  six   dimensions,   or   of  the    sixth 
degree. 

2 1 .  A  polynomial  is  said  to  be  homogeneous,  when  all 
its  terms  are  of  the  same  degree.     The  polynomial 

3a — 2Z»-j-c  is  of  the  first  degree  and  homogeneous. 

— 4ab-\-bz  is  of  the  second  degree  and  homogeneous. 

5a2c — 4c3-|-2c2d  is  of  the  third  degree  and  homogeneous. 

Sa3  -\-4ab-\-c  is  not  homogeneous. 

22.  A  vineulum  or  bar   ,  or  a  parenthesis  ( ), 

is  used  to  express  that  all  the  terms  of  a  polynomial  are  to 
be  considered  together.     Thus, 

a+b+cxb,   or    (o+5+c)x&, 

denotes  that  the  trinomial  a+b-\-c  is  to  be  multiplied  by  b  ; 
also,       a+b+cxc+d+f,   or    (a+6+c)  x  (c+d+/), 

denotes  that  the  trinomial   a+b+c   is  to  be  multiplied  by 
the  trinomial    c-\-d-\-f. 

When  the  parenthesis  is  used,  the  sign  of  multiplication 
is  usually  omitted.  Thus, 

(a-\-b+c)xb   is  the  same  as    (a-\-b  +  c)b. 

QWEST. — 2O.  What  is  the  dimension  of  a  term?  What  is  the  degree 
of  a  term  1  How  many  factors  in  3aic  1  Which  are  they!  What 
is  its  degree  1  21.  When  is  a  polynomial  homogeneous  1  Is  the  polyno- 
mial 2a3i4-3o2&2  homogeneous  1  Is  2a*6  —  is]  22.  For  what,  is  tb«» 
vineulum  or  bar  used  1  Can  you  express  the  same  with  the  parenthesis  7 


1  6  ELEMENTARY  ALGEBRA 

23.  The  terms  of  a  polynomial  which  are  composed  of 
the  same  letters,  the  same  letters  in  each  being  affected 
with  like  exponents,  are  called  similar  term?. 

Thus,  in  the  polynomial 


the  terms  7ab,  and  Sab,  are  similar  :  and  so  also  are  the 
terms  —  4a3b2  and  503Z>2,  the  letters  and  exponents  in  both 
being  the  same.  But  in  the  binomial  8a2b  -\-7ab2,  the 
terms  are  not  similar  ;  for,  although  they  are  composed  of 
the  same  letters,  yet  the  same  letters  are  not  affected  with 
like  exponents. 

24.  When    an   algebraic    expression   contains    similar 
terms,  it  may  be  reduced  to  a  simpler  form. 

1.  Take  the  expression    3ab+2ab,   which  is  evidently 
equal  to    5ab. 

2.  Reduce  the  expression  Sac  -\-9ac  -\-2ac  to  its  simplest 
form.  Ans.  I4ac. 

3.  Reduce  the  expression   abc-\-4abc-\-5abc  to   its  s'm- 
plest  form. 

In  adding  similar  terms  together  we  abc 

take  the   sum  of  the   coefficients   and  4abr 

annex  the  literal  part.     The  first  term,  5abc 

abc,    has    a    coefficient    1    understood,  lOabr. 
(Art.  12). 

25.  Of  the  different  terms  which  compose  a  polynomial, 
some  are  preceded  by  the  sign  -f-  ,  and  the  others  by  the 
sign  —  .     The  first  are  called  additive  terms,  the  others 
subtractive  terms. 


QUEST. — 23.  What  are  similar  terms  of  a  polynomial"?     Are 
and  6a2i2  similar  1     Are  2a2#2  and  2a3i2 1     24.  If  the  terms  are  positive 
and  similar,  may  they  be  reduced  to  a  simpler  form  7     In  what  wa.vv 


DEFINITION    OF    TERMS.  11 

The  first  term  of  a  polynomial  is  commonly  not  preceded 
by  any  sign,  but  then  it  is  understood  to  be  affected  with  the 
sign  +. 

1.  John   has    20    apples  and  gives  5  to  William:  how 
many  has  he  left  ? 

Now,  let  us  represent  the  number  of  apples  which  John 
has  by  «,  and  the  number  given  away  by  b  :  the  number  he 
would  have  left  would  then  be  represented  by  a—  b. 

2.  A  merchant  goes   into  trade  with  a  certain  sum  of 
money,  say   a   dollars  ;  at  the  end  of  a  certain  time  he  has 
gained   b   dollars  :  how  much  will  he  then  have  ? 

Ans.  a+b  dollars. 

If  instead  of  gaining  he  had  lost  b  dollars,  how  much 
would  he  have  had  ?  Ans.  a  —  b  dollars. 

Now,  if  the  losses  exceed  the  amount  with  which  he 
began  business,  that  is,  if  b  were  greater  than  a,  we  must 
prefix  the  minus  sign  to  the  remainder  to  show  that  the 
quantity  to  be  subtracted  was  the  greatest. 

Thus,  if  he  commenced  business  with  $2000,  and  lost 
$3000,  the  true  difference  would  be  —1000:  that  is,  the 
subtractive  quantity  exceeds  the  additive  by  $1000. 

3.  Let  a  merchant  call  the  debts  due  him  additive,  and 
the  debts  he  owes  subtractive.     Now,  if  he  has  due  him 
$600  from  one  man,  $800  from  another,  $300  from  another, 
and  owes  $500  to  one,  $200  to  a  second,  and  $50  to  a 
third,  how  will  the  account  stand  ?         Ans.  $950  due  him. 

4.  Reduce  to  its  simplest  form  the  expression 


QUEST.  —  25.  What  are  the  terms  called  which  are  preceded  by  the 
sign  +  ?  What  are  the  terms  called  which  are  preceded  by  the  sign  —  . 
If  no  sign  is  prefixed  to  a  term,  what  sign  is  understood  ?  If  some  of  the 
terms  are  additive  and  some  subtractive.  rnay  they  be  reduced  if  similar  ? 
Give  the  rule  for  reducing  them.  Does  the  reduction  affeet  the  expo- 
nents, or  only  the  coefficients  ? 

3 


18  ELEMENTARY  ALGEBRA. 

Additive  terms.  Subtractive  terms. 

-f 
-f 


_ 

Sum  +12a^  Sum  —  IQatb. 


But,  ,      I2azb— Wa2b  =  (. 

Hence,  for  the  reduction  of  the  similar  terms  of  a  polyno- 
mial we  have  the  following 

RULE. 

I.  Add  together  the  coefficients  of  all  the  additive  terms, 
and  annex  to  their  sum  the  literal  part ;  and  form  a  single 
subtractive  term  in  the  same  manner. 

II.  Then,  subtract  the  less  coefficient  from  the  gr  eater  * 
and  to  the  remainder  prefix  the  sign  of  the  greater  co- 
efficient,  to  which  annex  the  literal  part. 

REMARK. — It  should  be  observed  that  the  reduction  afle'cts 
only  coefficients,  and  not  the  exponents. 

EXAMPLES. 
I.  Reduce  to  its  simplest  form  the  polynomial 


Find  the  sum  of  the  additive  and  subtractive  terms  sepa- 
rately, and  take  their  difference  :  thus, 

Additive  terms.  Subtractive  terms. 


—  8a3bc* 
Sum  — 
Sum  -f-19a36c2 

IJence,  the  given  polynomial  reduces  to 


ADDITION. 

2.  Reduce  the  polynomial   4a26— Sazb— 9a26+lla26   to 
its  simplest  form.  Ans.  —2a2b. 

3.  Reduce   the   polynomial      7.abc2—abc2—7abc2  +  8abc2 
•f  6a£c2  to  its  simplest  form.  Ans.       I3abc2. 

4.  Reduce  the  polynomial  9cb3  —  8acz+15cb3+8ca  +  9ac* 
—  24cb3  to  its  simplest  form.  Ans.  ac2 -\-8ca. 

The  reduction  of  similar  terms  is  an  operation  peculiar  to 
algebra.  Such  reductions  are  constantly  made  in  Algebraic 
Addition,  Subtraction,  Multiplication,  and  Division. 


ADDITION. 

26.  Addition  in  Algebra,  consists  in  finding  the  simplest 
equivalent  expression  for  several  algebraic  quantities.  Such 
equivalent  expression  is  called  their  sum. 

1.  What  is  the  sum  of 

3ax+2ab   and    —  2ax+ab. 

3ax+2ab 

We  reduce  the  terms  as  in  Art.  25,         — 2ax+  ab 
and  find  for  the  sum ax+3ab 

(          3a 

2.  Let  it  be  required  to  add  together  \          ^ 

the  expressions  :  );' 


The  result   is 3<z+5£4-2< 


an  expression  which  cannot  be  reduced  to  a  more  simple 
form. 


QUEST. — 26.  What  is  addition  in  Algebra  ?     What  is  such  simplest 
and  equivalent  expression  called  ? 


20  ELEMENTARY  ALGEBRA. 


Again,  add  together  the   monomials  J          2a2b3 


The  result  after  reducing  (Art.  25),  is  .  .  I3a2b3 

r    2a2—4ab 
3.  Let  it  be  required  to  find  the  sum  \ 

of  the  expressions  , 


2ab-5b2 


Their  sum,  after  reducing  (Art.  25)  is  .  5a2—  Sab—  4b2 

27.  As  a  course  of  reasoning  similar  to  the  above 
would  apply  to  all  polynomials,  we  deduce  for  the  addition 
of  algebraic  quantities  the  following  general 

RULE. 

I.  Write  down  the  quantities  to  be  added  so  that  the  similar 
terms  shall  fall  under  each  other,  and  give  to  each  term  its 
proper  sign. 

II.  Reduce  the  similar  terms,  and  annex  to  the  results  the 
terms  which  cannot  be  reduced,  giving  to  each  term  its  respec- 
tive sign. 

EXAMPLES. 

1.  What  is  the  sum  of  Sax,  5ax,  —  2ax  and  I3ax.  ? 

Ans.  I9ax. 

2.  What  is  the  sum  of  4ab+8ac  and  2a£—  7ac+d.  ? 

Ans.  Gab  -\-ac-\-d. 

3.  Add  together  the  polynomials, 

3a2  —  2bz—  4ab,  5a2  —  b2+2ab,  and  Sab—  3c2—  252. 
The  term  3a2  being  similar  to 
5a2,   we  write    8az   for  the  result 

-  , 

of  the  reduction  of  these  two  terms, 

,.  ,  ,  . 

at  the  same  time  slightly  crossing 


^  .        -i        /. 

them,  as  in  the  first  term. 


_  2         ,        ,2 


--  —    O 

,2         _ 

—  2b*—  3cz 

—  —=-5  —  —  - 

o<2   -\-     CLO  —  OO   —  oC 


QUEST. — 27.  Give  the  rule  for  the  addition  of  Algebraic  quantities. 


ADDITION.  21 

Passing  then  to  the  term  —  4ab,  which  is  similar  to  -f  2ab 
and  +3ab,  the  three  reduce  to  -\~ab,  which  is  placed  after 
8a2,  and  the  terms  crossed  like  the  first  term  Passing 
then  to  the  terms  involving  b2,  we  find  their  sum  to  be 
— 5&2,  after  which  we  write  — 3c2. 

The  marks  are  drawn  across  the  terms,  that  none  of  them 
may  be  overlooked  and  omitted.  v 

(4)  (5)  (6) 

a  0«  5a 

a  5a  5b 

J2a  lla  5a+5b 

(9)  (10) 

7abc+9ax  8007+ 3b 

—3abc — 3ax  5ax — 9b 

•          4abc-\-§ax  j.3ax—6b 

NOTE. — If  a=5,  b=4,  c=2,  a?=l,  what  are  the  values 
of  the  several  sums  above  found. 

(12)  (13)  (14) 

9a-\-f  6ax —  8ac  3af-\-  g  -\-m 

— 6a-}-g  — 7ax —   9ac  ag — 3af — m 

— 2a—f  .     ax+\7ac  ab  — 

a-\-g  0           0  ab+4g 


(15)  (16) 

7a?+3a6+  3c 

-3x—3ab—  5c 

5x—  9ab —  9c  — 4x24-  4acx — 


9x— 9ab— lie  —  3#2-f-  0 


(17)  (18) 

22A— 3c— ' 


2ah2— 6a3b*+  ax3- 
3* 


22  ELEMENTARY  ALGEBRA. 

(19)''  (20) 

7x—  9y+5*+3—  g  8a+   b 

—  a  —  3y          —8—  #  2a—   6+   c 

—  a?+  y-3^+1  +  7^  —  3a+   fc 

—  2x+6y+3z—  1  —  ^  —66—  3c-f3d 

—  5a         +7c—  8<f 


4a?H-3y+0  +4  +  5^  2a  —  5b+5c—  3d 


21.  Add  together  —  6+3c—  ^—  115e+6/-5^,  36-2c 
3rf__C4_27/,  5c—  8rf+3/—  7^,  —  7i—  6c+17<Z+9e—  5/ 
,  —  3J—  5d—  2e+6/— 


22.  Add  together  the  polynomials   7a2b—  3abc—  8b2c—  9c3 

and    4a25—  8c3-f952c 
—  352c—  14c3-f  2cd2—  3d3. 

23.  What  is  the  sum  of  5a25c+65a:—  4af,   —3a2bc—6bx 
af,    —af+9bx+2a2bc,    +6af—8bx+6a?bc. 

Ans.   Watbc+bx+lSaf. 

24.  What  is  the  sum  of  «2n2+3a3m-f  b,  —  6a2w2—  6a3m—  b, 


25.  What  is  the  sum  of  4a3b2c—16a*x—9ax3d,  +6a3b2c 
—Gaxtd+Wtfx,    -\-l6ax3d—a*x—9a3b2c. 

Ans.  a3bzc-\-ax3d. 

26.  What   is   the    sum   of    —7^+3^4-4^—2^     +3# 

—  3b-i  2b.  Ans.  0. 

27.  What  is  the  sum  of   ab-\-3xy  —  m  —  n,    —  6xy  —  3m 
+  lln+cd,    +3iry  +  4m—  Wn+fg.  Ans.  ab  +  cd+fg. 

28.  What  is  the  sum  of  4xy+n  -\-6ax-\-9am,  —  6xy+6n 

—  6ax—8am,   2xy—7n+ax—am.  Ans.  +ax. 


SUBTRACTION.  23 


29.  Add  the  polynomials  IQaWb—  I2a3cb, 

—  Wax,    —2a2x3b—l2a*cb,  and  -—  I8a2x3b—  12a3cb+9ax. 

Ans.  4a?x3b—22a3cb—ax. 

30.  Add   together    3a-f-£+c,     5a+2b  +  3ac,    a+c+ac, 
and  —  3a—  9ac  —  8b.  Ans.  6a  —  55+2c—  5ac. 

31.  Add  together     .     5a?b  +  6cx+9bc*,    7cx—8a?b,    and 

—  15cx—  9bc2+2a2b.  Ans.   —a2b  —  2cx. 

32.  Add  together  8a#+5a&-f  3a252c2,  —  18aa;-f6a2-f-10a& 
and  10a#—  I5ab  —  6a2bzc2.  Ans.   —  3a262c2-f  6a2. 

33.  Add  together  3a2+5a2J2c2—  9a3#,  7a2  —  8a252c2—10a3» 
and 


SUBTRACTION. 

28.  Subtraction,  in  Algebra,  consists  in  finding  the  sim- 
plest expression  for  the  difference  between  two  algebraic 
quantities. 

Thus,  the  difference  between  6a  and  3<z  is  expressed  by 

60—  3«=3a; 
and  the  difference  between  7a?b  and  3a3&  by 


In  like  manner  the  difference  between  4a  and  3b  is 
expressed  by  4a—  3b. 

Hence,  If  the  quantities  are  positive  and  similar,  subtract 
the  coefficients,  and  to  their  difference  annex  the  literal  part. 
If  they  are  not  similar,  place  the  minus  sign  before  the  quantity 
to  be  subtracted. 

QUEST.  —  28.  In  what  does  subtraction  in  Algebra  consist  ?  How  do 
you  find  this  difference  when  the  quantities  are  positive  and  similar  ? 
When  they  are  not  similar,  how  do  you  express  the  difference  ? 


ELEMENTARY  ALGEBRA. 


From 

take 

Rem. 


From 

take 

Rem. 


From 

take 

Rem. 


(1) 

3ab 

2ab 

ab 

(4) 


(3) 

9abc 
Tabc 


(6) 


(9) 
2am 

ax 


3ax—8c 


4abx — 9ac 


2am — ax. 


29.  Let  it  be  required  to  subtract  from     4a 
the  binomial  .  2b  —  3c 


The  difference  may  be  put  under  the  form  4a  —  (2b  —  3c). 
We  must  now  remark  that  it  is  the  difference  between  2b 
and  3c  which  is  to  be  taken  from  4a. 

If  then,  we  write 4a — 2b, 

we  shall  have  taken  away  too  much  by  the  units  in  3c ; 
hence,  3c  must  be  added  to  give  the  true  remainder,  .which 
is 4a— 26  +  3c. 

To  illustrate  this    example   by  figures,   suppose    a =5, 
b=5t   and   c=3. 

We  shall  then  have 4a=20 

and .    2b— 3c  -10—9   —   1 

which  may  be  written       .     4a  —  (2b  —  3c)—20  — 1"=  19. 


QUEST. — 29.  If  2i  —  3c  is  to  be  taken  from  4a,  what  is  proposed  to 
be  done  ?  If  you  subtract  26  from  4a,  have  you  taken  too  much  ?  /low 
then  must  you  supply  the  deficiency  ? 


SUBTRACTION.  25 

Here  it  is  required  to  subtract  1  from  20.  If,  then,  we 
subtract  2b=lO,  from  4a  =  20,  it  is  plain  that  we  shall 
have  taken  too  much  by  3c=9,  which  must  therefore  be 
added  to  give  the  true  remainder. 

3O.  Hence,  for  the  subtraction  of  algebraic  quantities 
«re  have  the  following  general 

RULE. 

I.  Write  the  quantity  to  be  subtracted  under  that  from  which 
it  is  to  be  taken,  placing  the  similar  terms,  if  there  are  any, 
under  each  other. 

II.  Change  the  signs  of  all  the  terms  of  the  polynomial  to 
be  subtracted,  or  conceive  them  to  be  changed,  and  then  reduce 
the  polynomial  result  to  its  simplest  form. 

EXAMPLES. 


From 
Take 
Rem. 

From 
Take 
Rem. 

From 
Take 
Rem. 

(i) 

6ac  —  5ab-{-   c2 
3ac-\-3ab-\-7c 

The  same  with 
the  signs  of  the 
lower  line  chan- 

(1) 

6ac—  5ab+   c2 
—  Sac  —  3ab  —  7c 

3ac—8ab+   c2—  7c. 

1        Sac—  8ab+   cz—7c. 

(2) 
6ax—  a-f  S52 
9ax—  x+   b2 

(3) 
6yx—  3x2+5b 
^yx-T-3    -{-  a 

—  3ax—a-\-x-\-2b2. 

5yx—  3«2+3  +  5&—  a. 

(4) 

5a3_4a2£_|_     3£2C 

—  2a3+3a2£  —   Sb2c 

(M 

4ab—  cd-\-3a2 
5ab—4cd+3a2+5b2 

7a*—7a2b+nb2c. 

—  ab+3cd—5b2. 

QUKST. — 30.  Give  the  rule  for  the  subtraction  of  algebraic  quantities. 


26  ELEMENTARY  ALGEBRA. 

6.  From   6am-\-y   take    Sam — x.  Ans.  3am-\-x+y. 

7.  From   Sax   take    3>ax — y.  Ans.  -j-y 

8.  From   7aW—x*   take    I8a?bz+xz. 

Ans.   —  llaW—2x2. 

9.  From    —7f+3m—8x   take    — 6/—  5m— 2x+3d+8. 

Ans.   —f+8m—6x—3d—8 

10.  From    —  a— 5b+7c—d  take    45— c  +  2d+2A:. 

Ans.   —a—9b  +  8c—3d—2k 

11.  From  .  .    —  3a+b— 8c+7e— 5/+3A— 7x— 13y   take 
*+2a— 9c+8e— 7a?-f-7/—  y— 3Z— A. 

^.n^.   —  5a-f£-f  c— e  —  12/+3A— 12y+3/. 

12.  From   a+b  take    a— b.  Ans.  2b. 

13.  From   2x— 4a— 2^+5   take    8— 5&-fa-f 6#. 

^.n^.  — 4a?— 5a+35— 3 

14.  From   3a+J+c— rf—10   take    c-f  2a— d. 

^.n^.  a-\-b— 10 

15.  From   3a+5+c— d— 10   take    5— 19+3a. 

Aw5.  c— 

16.  From   2a&+52— 4c+5c— b   take    3a2— 

^.ns.  2ai— 3a2— 

17.  From   a*+3bzc+abz—abc   take    b3+ab2—abc. 

Ans.  a3+3£2c— R 

18.  From    12a?+6a— 4&+40  take   4b— 3a+4x  +  6d— 10. 

Ans.  8;r-f9a— 8b  —  6c?+50. 

19.  From  2x— 3a+454-6c— 50  take  9a+x+6b  —  6c— 40. 

Ans.  x— 12a— 2£-f-12c— 10. 

20.  From    6a—4i— 12c+l2x   take    2o;— 8a+4*— 6c. 

^.n^.  14a — 8b— 6c+10a?. 

21.  From  8«5c— 12b3a+6cx— 7xy  take   7c«— xy—13b3a. 

A  ns.  8abc  -f-  b*a  —  ex — 6xy 


SUBTRACTION.  27 

31.  By  the  rule  for  subtraction,  polynomials  may  be 
subjected  to  certain  transformations. 

For  example,     .     .  6a2—  Sab  -f  252  —  2bc, 

becomes 6a2— (3ab  —  2£2 

En  like  manner      .     .  7a3—  8azb— 

becomes       ....  7a3— (8a26  +  462c— 6&2), 

or,  again,      ....  7a3—  8a2£— (462c— G62). 

Also, 8a3—  7£3  +     c    —  d, 

becomes       ....  8<z3— (7Z>2  —    c    +d). 

Also, 963—     a     +  3a2  —  d, 

becomes       ....  9b3  —  (a     —  3a2  -f  d). 

32.  REMARK. — From  what  has  been  shown  in  addition 
and  subtraction,  we  deduce  the  following  principles. 

1st.  In  algebra,  the  words  add  and  sum  do  not  always,  as 
m  arithmetic,  convey  the  idea  of  augmentation.  For,  if  to  a 
we  add  — 6,  we  have  a — &,  which  is,  properly  speaking,  a 
difference  between  the  number  of  units  expressed  by  «,  and 
the  number  of  units  expressed  by  b.  Consequently,  this 
result  is  numerically  less  than  a.  To  distinguish  this  sum 
from  in  arithmetical  sum,  it  is  called  the  algebraic  sum.  p27 

Thus,  the  polynomial  2a2 — 3azb-\-3b2c '  is  an  algebraic 
sum,  so  long  as  it  is  considered  as  the  result  of  the  union 


QUEST. — 31.  How  may  you  change  the  /orm  of  a  polynomial  ?  32.  In 
algebra  do  the  words  add  and  sum  convey  the  same  idea  a"s  in  arithme- 
tic ?  What  is  the  algebraic  sum  of  9  and/— 4?  Of  8  and  — 2? 
May  an  algebraic  sum  ever  be  negative  ?  What  is  the  sum  of  4  and 
• — 8"?  Do  the  words  subtraction  and  difference  in  algebra  always  con- 
vey the  idea  of  diminution  ?  What  is  the  algebraic  difference  between 
8  and  — 4?  Between  a  and  — bl 


28  ELEMENTARY  ALGEBRA 

of  the  monomials   2a2,  —3a2b,  -f-352c,  with  their  respec 
tive  signs  ;  and,  in  its  proper  acceptation,  it  is  the  arithmeti- 
cal difference  between  the  sum  of  jthe  units  contained  in  the 
additive  terms,  and  the  sum  of  the  units  contained  in  the 
subtractive  terms. 

It  follows  from  this,  that  an  algebraic  sum  may,  in  the 
numerical  applications,  be  reduced  to  a  negative  number,  01 
a  number  affected  with  the  sign  — . 

2nd.  The  w«ords  subtraction  and  difference  do  not  always 
convey  the  idea  of  diminution ;  for,  the  numerical  differenca 
between  -{-a  and  — b  being  a-{-b,  exceeds  a.  This  result 
is  an  algebraic  difference,  and  can  be  put  under  the  form  of 

a_  (—b)= a+b 


MULTIPLICATION. 

33.  If  a  man  earns  a  dollars  in  one  day,  how  much 
will  he  earn  in  6  days  ?  Here  it  is  simply  required  to  re- 
peat the  number  a,  6  times,  which  gives  6a  for  the  amount 
earned. 

1.  What  will  ten  yards  of  cloth  cost  at  c  dollars  per  yard? 

Ans.   lOc  dollars. 

2.  What  will  d  hats  cost  at  9  dollars  per  hat  1 

Ans.  9d  dollars. 

3.  What  will  b  cravats  cost  at  40  cents  each  ? 

Ans.  40b  cents. 

4.  What  will  b  pair  of  gloves  cost  at  a  cents  a  pair  ? 


QUEST. — 33.  What  is  the  object  of  multiplication  in  algebra  ?  It  a 
man  earns  a  dollars  in  one  day,  how  much  will  ho.  earn  in  4  days'?  In 
5  days  1  In  6  days  * 


MULTIPLICATION.  29 

Here  it  is  plain  that  the  cost  will  be  found  by  repeating  b 
as  many  times  as  there  are  units  in  a  :  Hence,  the  cost  is 
ab  cents.  Ans.  ab  cents. 

NOTE.  —  If  we  suppose  a=6,  c=4,  and  d=3t  wha-t  would 
be  the  numerical  values  of  the  above  answers  ? 

5.  If  a  man's  income  is  3a  dollars  a  week,  how  much 
will  it  be  in  4b  weeks.  Here  we  must  repeat  3a  dollars  as 
many  times  as  there  are  units  in  4b  weeks  ;  hence,  the  pro- 
duct is  equal  to 


If  we  suppose  a=4=  and  b  =  3  the  product  will  be  equal 
to  144. 

34.  REMARK.  —  It  is  plain  that  the  product  I2ab  will  not 
be  altered  by  changing  the  arrangement  of  the  factors  ;  that 
is,  I2ab   is    the    same    as    a6x!2,   or   as   fozx!2,   or   as 
axl2xb  (See  Arithmetic,  §  22). 

35.  Let  us  now  multiply  3a2b2  by  2a2b,  which  may  be 
placed  under  the  form 

3a2b2  X  2a2b=3  X2aaaabbb  ; 

in  which  a  is  a  factor  four  times,  and  b  a  factor  three  times  : 
hence  (Art.  13). 


in  which,  we  multiply  the  co-efficients  together  and  add  the 
exponents  of  the  like  letters. 


QUEST. — 34.  Will  a  product  be  altered  by  changing  the  arrangement 
if  the  factors  1  Is  3ai  the  same  as  3ba  1  Is  it  the  same  as  a  X  3i  t 
As  5x3a  1  35.  In  multiplying  monomials  what  do  you  do  with  the  co- 
efficients 1  What  do  you  do  with  the  exponents  of  the  common  letters  ? 
If  a  letter  is  found  in  one  factor  and  not  in  the  other,  what  do  you  do  1 

4 


80  ELEMENTARY  ALGEBRA. 

Hence,  for  the  multiplication  of  monomials,  we  have  the 
following 

RULE. 

I.  Multiply  the  coefficients  together  for  a  new  coefficient. 

II.  Write  after  this  coefficient  all  the  letters  which  enter 
into  the  multiplicand  and  multiplier^  affecting  each  with 
an  exponent  equal  to  the  sum  of  its  exponents  in  both  fac- 

tors. 

• 

EXAMPLES. 


2.  ZltfVcdx  8ab<?=  168oW</. 

3.  4abcX7df  =  VSabcdf. 


(4) 

(5) 

(6; 

Multiply 

3a2b 

I2a2x 

6a:y  z 

by 

2a2b 

I2x2y 

ay2z 

6a4b* 

I44a2x3y 

6axy3zz 

(7) 

(8) 

(9) 

LtX 

87«o;2y 

2a?x2y 


10.  Multiply    5a3bW  by  6c5x6.  Ans. 

11.  Multiply  10«465c8  by  7acd.  Ans.  7Qa5b5c*d. 

12.  Multiply  9a3bxy  by  9a3bxy.  Ans. 

13.  Multiply  36a867c6d5  by  2Qab2c3d*.  Ans. 

14.  Multiply  27axyz  by  9a2b2c2d2xyz. 

Ans. 

15.  Multiply  13«3&2c  by  Sabxy.  Ans    \Q4a*b3cxy 


MULTIPLICATION.  31 


16.  Multiply  2Qcfb5cd  by  12aVfy.       dns. 

17.  Multiply  l4aWy  by  2Qa5c2x2y. 

tins. 

18.  Multiply  Scfby  by  7a*bxy*.  Ans.  56a7&4a*/9. 

19.  Multiply  75axyz  by  5a5bcdx2y\ 

tins.  275a?bcdxsysz. 

20.  Multiply  Slayer2  by  §a*bc*y?y.       ./2/is.  459a46cVy. 

21.  Multiply  2as&y  by  ISabx.    *  .tfrcs. 

22.  Multiply  64aVx4yz  by 


23.  Multiply  9a2#W3  by  12aW.          .flns. 

24.  Multiply  216a6W  by  3a362c5.         ^ns.  648aWd8. 

25.  Multiply  70a8b7c*d2fx  by  1 


36.  We  will  now  consider  the  most  general  case  of  two 
polynomials. 

Let  a  represent  the  sum  of  all  the  additive  terms  of  the 
multiplicand,  and  b  the  sum  of  the  subtractive  terms.  Let  c 
denote  the  sum  of  the  additive  terms  of  the  multiplier,  and  d 
the  sum  of  the  subtractive  terms.  The  multiplicand  will 
then  be  represented  by  a  —  6,  and  the  multiplier  by  c  —  d;  and 
it  is  required  to  take  a  —  b  as  many  times  as  there  are  units  in 
c—  d. 

Let  us  first  take  a  —  b  as  many  times         a  —  b 
as  there  are  units  in  c.  c  —  d 

We  begin  by  writing  ac,  which  is         ac—lc 
too  great  by  b  taken  c  times;  for  it  is  _  ad-\-hd 

only  the  difference  between  a  and  b         ac—bc—ad+bd. 
which  is  to  be  taken  c  times.     Hence, 
ac  —  be  is  the  product  of  a  —  b  by  c. 

But  it  was  proposed  to  take  a  —  b  only  as  many  times  as 
there  are  units  in  the  difference  between  c  and-c/:  hence  the 


32  ELEMENTARY  ALGEBRA. 

last  product  ac  —  be  is  too  large  by  a  —  b  taken  d  times  ;  which 
without  regarding  the  sign  of  d,  is  ad  —  bd.  Changing  the 
signs  of  this  last  product,  and  subtracting  it  from  that  of  a  —  b 
J)y  c  (Art.  3O),  and  we  have 

(a—  b)  x  (c—  d)  =  ac  —  be  —  ad+  bd. 
~  37.  Hence,  we  have  the  following  rule  for  the  signs. 

When  two  terms  of  the  multiplicand  and  multiplier  are 
affected  with  the  same  sign,  the  corresponding  product  is 
affected  with  the  sign  +;  and  when  they  are  affected  with 
contrary  signs,  the  product  is  affected  with  the  sign  —  . 

Therefore  we  say  in  algebraic  language,  that  -f  multiplied 
by  H-  ,  or  —  multiplied  by  —  ,  gives  +  ;  —  multiplied  by 
+  ,  or  -f  multiplied  by  —  ,  gives  —  . 

Hence,  for  the  multiplication  of  polynomials  we  have  the 
following 

RULE. 

Multiply  all  the  terms  of  the  multiplicand  by  each  term  of 
the  multiplier,  observing  that  like  signs  give  plus  in  the  pro- 
duct, and  unlike  signs  minus.  Then  reduce  the  polynomial 
result  to  its  simplest  form. 

EXAMPLES  IN  WHICH  ALL  THE  TERMS  ARE   PLUS. 


1  Multiply     ...... 

by      .......       20  +  56 


The  product,  after  reducing,       .  +l5azb+2Qab2-\-5b3 

becomes       .....       Go3  -f  23a2b  +  22ab2  +  563. 


QUEST.  —  37.  What  does  -f  multiplied  by  -f  give  ?  -f  multiplied 
by  —  ?  —  multiplied  by  -f  ?  —  multiplied  by  —  ?  Give  the  rule  for 
the  multiplication  of  polynomials. 


MULTIPLICATION.  33 

2.  Multiply  z?+2ax+  a2  by  x+a. 

Jlns.  ar}+3a#2+3a2;r-f-as. 

3.  Multiply  x3+y3  by  x+y.    -     *fl.ns.  x4-\-xy3+xsy-i-y*. 

4.  Multiply  3ab*+6tf<*  by  3a62+3o2c2. 


5.  Multiply  tftf  +  fd  by  a  +b. 

Ans. 

6.  Multiply  3ax*+9abs+cd5  by  6aV 


7.  Multiply  64aV+27a2#+9a&  ly  8a3cJ 

rfns.  Sltofcd 

8.  Multiply  aHSa^+or2  by  a  +  x. 

Jlns. 

9.  Multiply  (f  +  Scfx+Satf+x3  by 


10.  Multiply  a^+y2  by  x+y.      tins.  xs+xy2+x2y+y3. 

11.  Multiply  x*-\-xy6-\-7ax  by 


12.  Multiply  a3+3a26  +  3«62  +  63  by  a  +b. 

Jlns.  a*+ 

13.  Multiply  x3-}-x2y+xif+y3  by 


14.  Multiply  a;3+2a;2  +  a;  +  3  by  3a*+l. 
Ans. 


GENERAL  EXAMPLES. 

1.  Multiply 20#  —  Sab 

by        3x    —     b. 

The  product Gax2—  9abx 

becomes  after —  2«&r-f  3a52 

reducing Gax2—  llabx+3ab2. 


34  ELEMENTARY.  ALGEBRA. 

2.  Multiply   a4—  2b3   by    a—  b. 

Ans.  a5—2ab3— 

3.  Multiply   x2—  3x—  7   by   x—2. 

Ans.  x3—  5x2—  a?-J-14 

4.  Multiply    3a2  —  5ab  +  2b2   by   a2  —  7ab. 

Ans.  3a*—26a3b+37a2b2  —  Uab* 
6.  Multiply   &2+&4+&6   by   £2—  1.  Ans.  b8—b2 

6.  Multiply   x*—2x3y  +  4x2y2  —  8xy3+16y*   by   #+2y. 

Arcs.  rr5+32y5. 

7.  Multiply   4a:2—  2y   by   2y.  Ans,  8x*y  —  4y2. 

8.  Multiply   2x+4y   by   2x—  4y.  Ans.  4x2—l6yz. 

9.  Multiply   x3-\-x2y-\-xy2-}-y3   by   #—  y. 

-An^.  a?4  —  y4. 

10.  Multiply   ff2  +  #y+y2   by   x2—xy+y2. 

Ans.  #4-f-#2y2-|-y4. 

11.  Multiply   2az—3ax+4x*   by    5a2—  6^-2«2. 

Ans.   10a4—  27a3x+34azx2—  I8ax3—  8x*. 

12.  Multiply    3x2—  2a?y45    by   a:2+2a:y—  3. 

An^.  3*4+4a;3y—  4a;2  —  4a:2y2+16a:y—  15. 

13.  Multiply    3tf34-2a:2y2-f  3y2    by    2x3  —  3a;2y2 


'  C 

14.  Multiply    Sax—  Gab—  c   by 

Ans.   1  6a2#2  —  4a2  bx  —  6azbz  -\-  6acx  —  7abc  —  c2. 

15.  Multiply    3a2—  552  +  3c2   by   a2  —  b2. 

Ans.  3a4  — 

16.  3a2  —  55d!+   c/ 

—  8cf. 


Pro.  red.    ^-15a4+37a2^—  29a?cf— 


MULTIPLICATION.  35 

38.  To  finish  with  what  has  reference  to  algebraic  mul- 
tiplication, we  will  make  known  a  few  results  of  frequent 
use  in  Algebra. 

Let  it  be  required  to  form  the  square  or  second  power  of 
the  binomial  (a-f-Z>).  We  have,  from  known  principles, 


That  is,  the  square  of  the  sum  of  two  quantities  is  equal  to 
the  square  of  'the  first  ,  plus  twice  the  product  of  the  first  by  the 
second,  plus  the  square  of  the  second. 

1.  Form  the  square  of  2a+3b.     We  have  from  the  rule 

(2a  -f  36)2      =   4a2    +    I2ab    -f   952. 

2.  (5ab+3ac)2     =  25aW+   30a*bc+   9a2c2. 

3.  (5a2+8a26)2  =25a*     +    80a4b 

4.  (6aa;-f9«2a;2)2  =  36a2a;2+108a3a:3 

39.  To  form  the  square  of  a  difference   a  —  &,  we  have 


That  is,  the  square  of  the  difference  between  two  quantities  is 
equal  to  the  square  of  the  first,  minus  twice  the  product  of  the 
first  by  the  second,  plus  the  square  of  the  second. 
1.  Form  the  square  of  2a—b.     We  have 


2.  Form  the  square  of  4ac  —  be.     We  have 

(4ac  —  6c)2  =  16a2c2—  8abcz+bzcz. 

3.  Form  the  square  of  7azb2  —  I2ab*.     We  have 


QUEST. — 38.  What  is  the  square  of  the  sum  of  two  quantities  equal  to  ? 
39.  What  is  the  square  of  the  difference  of  two  quantities  equal  to  ? 


36  ELEMENTARY  ALGEBRA. 

4O.  Let  it  be  required  to  multiply  a-\-b  by  a  —  b.     We 
have 


Hence,  the  sum  of  two  quantities,  multiplied  by  their  differ* 
ence,  is  equal  to  the  difference  of  their  square* 

1.  Multiply   2c+5   by   2c—  b.     We  have 

(2c-f&)x(2c-&)z=4c2-£2. 

2.  Multiply   9<zc-f-35c   by   Qac—  3bc.     We  have 


3.  Multiply   8a3+7ab2   by   8a3—7ab2.     We  have 
(8a3+7a&2)(8a3—  7a62)  =  64a6—  49a2Z>4. 


41.  It  is  sometimes  convenient  to  find  the  factors  of  a 
polynomial,  or  to  resolve  a  polynomial  into  its  factors. 
Thus,  if  we  have  the  polynomial 

ac-\-ab  -\-adj 

we  see  that   a   is  a  common  factor  to  each  of  the  terms  : 
hence,  it  may  be  placed  under  the  form 

a(c+b+d). 

1.  Find  the  factors  of  the  polynomial   azb2+azd-{-a2f. 

Ans. 

2.  Find  the  factors  of  3a2b+6a2b*+b2d. 

Ans.  b(Z 

3.  Find  the  factors  of  3<z2&+9a2c+18a2o:y. 

Ans. 


QUEST. — 40.  What  is  the  sum  of  two  quantities  multiplied  by  theii 
difference  equal  to! 


DIVISION.  37 

4.  Find  the  factors  of   8a*cx—  18acx2+2ac5y— 30a6c9a?. 

Ans. 

5.  Find  the  factors  of  a2-\ 

Ans.  (a+b)x(a+b). 

6.  Find  the  factors  of   a2  — b*.          Ans.  (a+b)x(a—b). 

7.  Find  the  factors  of  a2— 2ab+bz. 

Ans.  (a—b)x(a—b). 


DIVISION. 

42.  Algebraic  division  has  the  same  object  as  arithmeti- 
cal, viz  :  having  given  a  product,  and  one  of  its  factors,  to 
find  the  other  factor. 

We  will  first  consider  the  case  of  two  monomials. 

The  division  of  72a5   by    8a3   is  indicated  thus  : 

72a5 
W 

It  is  required  to  find  a  third  monomial,  which,  multiplied 
by  the  second,  will  produce  the  first.  It  is  plain  that  the 
tiiird  monomial  is  9a2  ;  for  by  the  rules  of  multiplication 


72  a5 

Hence,  we  have  •  •      =9a2, 

8aJ 

a  result  which  is  obtained  by  dividing  the  coefficient  of  the 
dividend  by  the  coefficient  of  the  divisor,  and  subtracting  the 
exponents  of  the  like  letter. 

QUEST.  —  42.  What  .is  the  object  of  division  in  Algebra]     Give  the 
rule  for  dividing  monomials! 


38 


ELEMENTARY  ALGEBRA. 


Also, 


Tab 
for,  7ab  X  5a2bc  = 


Hence,  lor  the  division  of  monomials  we  have  the  following 


I.  Divide  the  coefficient  of  the  dividend  by  the  coefficient 
of  the  divisor,  for  a  new  coefficient. 

II.  Write  after  this  coefficient,  all  the  letters  of  the  divi- 
dend, and  affect  each  with   an  exponent  equal  to  the  ex- 
cess of  its    exponent   in  the  dividend  over  that   in   the 
divisor. 


From  these  rules  we  find 
48oWd 


1.  Divide  Wx2  by  Sx. 

2.  Divide  15a*y  by  Say. 

3.  Divide  *84ab3x   by    I2b2. 

4.  Divide  36a465c2   by    9a3Z>2c. 

5.  Divide  88a3b2c   by    8a2b. 

6.  Divide  99a*b*x5   by    Ila352a;4. 

7.  Divide  108a?6y5^3   by   54x5z. 

8.  Divide  64x7y5^6   by    I6x6y*z5. 

9.  Divide  96a756c5   by    12a25c. 

10.  Divide  54a7c5^6   by    27acd. 

11.  Divide  38a466d4   by   2a3b5d. 


£ns. 

dns. 
Ans.  7abx. 

Ans.  4ab3c. 

Ans.  llabc. 

Ans.  9ab2x 
Ans.  2xy5z2. 

Ans.  ±xyz. 
Ans.  8a5b5c*. 
Ans.  2a6c*d5. 
Ans.  I9abd*. 


DIVISION.  39 

12.  Divide  42a262c2   by    7abc.  Ans.  6abc. 

13.  Divide  64a564c8   by    32a*bc.  ^n.?.  2a£3c7. 

14.  Divide  128a5z6y7    by    IQaxy*.  Ans.  8a4tf5y3. 

15.  Divide  132W5/6   by   2d4/.  Ans.  66 bdf5. 

16.  Divide  256a46W7    by    I6a3bc6.  Ans.  16ab8c2<F 

17.  Divide  200a8m2n2   by    dOtfmn.  Ans.  4amn. 

18.  Divide  300^3y%2   by    60^y%.  Ans.  5x2y2z. 

19.  Divide  27a5£2c2   by    9a6c.  An*.  3a45c. 

20.  Divide  64a3y6^8   by    32ay5z1.  Ans.  2azyz. 

21.  Divide  88a566c8   by    Ila3&4c6.  An5.  8fl26^. 

43.  It  follows  from  the  preceding  rule,  that  the  division 
of  monomials  will  be  impossible, 

1st.  When  the  coefficients  are  not  divisible  by  each  other. 

2nd.  When  the  exponent  of  the  same  letter  is  greater 
in  the  divisor  than  in  the  dividend. 

3rd.  When  the  divisor  contains  one  or  more  letters  which 
are  not  found  in  the  dividend. 

When  either  of  these  three  cases  occurs,  the  quotient  re- 
mains under  the  form  of  a  monomial  fraction ;  that  is,  a 
monomial  expression,  necessarily  containing  the  algebraic 
sign  of  division,  but  which  may  frequently  be  reduced. 

Take,  for  example,  12a4&2cc?,  to  be  divided  by  8a2bc2. 
which  is  placed  under  the  form 

\2aWcd 


QUEST. — 43.  What  is  the  first  case  named  in  which  the  division  of 
jflonomials  will  not  be  exact  1  What  is  the  second  1  What  is  the  third  1 
If  either  of  these  cases  occur,  can  the  exact  division  be  made1?  Under 
what  form  will  the  quotient  then  remain  1  Mav  this  fraction  be  often 
reduced  to  a  simpler  form  1 


2c 
5a 


40  ELEMENTARY  ALGEBRA. 

this  may  be  reduced  by  dividing  the  numerator  and  denomi- 
nator by  the  common  factors  4,  a2,  b,  and  c,  which  gives 

3a*bd 


Also, 


44.  Hence,  for  the  reduction  of  a  monomial  fraction  to  its 
simplest  form,  we  have  the  following 

RULE. 

Suppress  every  factor,  whether  numerical  or  literal,  that 
is  common  to  both  terms  of  the  fraction,  and  the  result  will 
be  the  reduced  fraction  sought.  „ 

From  this  new  rule,  we  find, 

(1)  (2) 

370 


=— T: —  :     and 


36a*b3czde~~  3bce 

(3)  (4) 


also        .'"^    =  0\  :     and    — ^^ — = — — 
2ao  bao*  oo 


7bc2 
5.  Divide    49a262c6   by    14a3ic4.  Ans. 


6.  Divide    Gamra   by   3abc.  .  Ans. 

7.  Divide    18a252mn2   by    12a*Mcrf.  An*. 


2a 
2mn 


QUEST.  —  44.  Give  the  rule  for  the  reduction  of  a  monomial  fraction. 


DIVISION.  41 

8.  Divide    28<75#5c7<Z8   by    l6ab9ccFm.  Ans. 


6 
9    Divide    72a3c2b2   by    I2a5c*b3d.  Ans. 


10.  Divide    I00a8b5xmn   by   25a3b4d.        Ans.  -^—  — 


32a3b*csdf 

11.  Divide    96a5b*c*df  by    75a2cxy.          Ans.  J-  . 

1  7n2x2v3 

12.  Divide    85m2w3/x2y3   by    15aw4n/.       Ans.    —  —  f-  . 

127 

13.  Divide    I27d3x2y2    by    16c/4#V-  Ans-  T^ 


45.  If  we  have  an  expression  of  the  form 

a  a2  a3  a4  a5 

—  ,     or     —  ,     or     —  ,     or     —  ,     or     —  , 
a  a2  a3  a4  a* 

and  apply  the  rule  for  the  exponents,  we  shall  have 


But  since  any  quantity  divided  by  itself  is  equal  to   1,  it 
follows  that 

—  =aO=i      fl=aa-a=tfo==i    &Cj 

a  a2 

>r  finally,  if  we  designate  the  general  exponent  by  m,  we 
have 

—=am-m=a«=l  ; 
«"» 

that  is,  any  /?0MJer  o/"  which  the  exponent  is  0  is  equal  to  1;  and 
hence,  a  factor  of  the  form  a°,  being  equal  to  1,  may  be  omitted. 

QUEST.—  45.  What  is  a°  equal  to  ?     What  is  6°  equal  to  ?     What  is 
the  power  of  any  number  equal  to,  wfyeu  tl»e  exponent  of  the  power  is  0  ? 


42  ELEMENTARY  ALGEBRA. 

2.  Divide    6a?b2c*d   by   2azbzd. 


2azbzd 


3.  Divide    So^W   by   4a258c*<P.  An*.  2a2 

4.  Divide  16a6W   by    8aW.  Ans.  2<28. 

5.  Divide    32m3n3oc'2yz   by    4m3n3xy.  Ans.  8,ry. 

6.  Divide    96a4£5d8c9   by   24aWd5c*.  Ans.  4bd3 

SIGNS  IN  DIVISION. 

4G.  The  object  of  division,  is  to  find  a  third  quantity 
called  the  quotient,  which,  multiplied  by  the  divisor,  shall 
produce  the  dividend. 

Since,  in  multiplication,  the  product  of  two  terms  having 
the  same  sign  is  affected  with  the  sign  -f,  and  the  product 
of  two  terms  having  contrary  signs  is  affected  with  the 
sign  — ,  we  may  conclude, 

1st.  That  when  the  term  of  the  dividend  has  the  sign  -f, 
and  that  of  the  divisor  the  sign  of  -f-j  the  term  of  the  quo- 
tient must  have  the  sign  ~f-  • 

2nd.  When  the  term  of  the  dividend  has  the  sign  -\- ,  and 
that  of  the  divisor  the  sign  — ,  the  term  of  the  quotient 
must  have  the  sign  — ,  because  it  is  only  the  sign  — , 
which,  multiplied  with  the  sign  — ,  can  produce  the  sign  -f- 
of  the  dividend. 


Q0EST. — 46.  What  will  the  quotient,  multiplied  by  the  divisor,  be 
equal  to  ?  If  the  multiplicand  and  multiplier  have  like  signs,  what  will 
be  the  sign  of-  the  product?  If  they  have  contrary  signs,  what  will  be 
the  sign  of  the  product?  When  the  term  of  the  dividend  and  the  term 
of  the  divisor  have  the  same  sign,  what  will  be  the  sign  of  the  quotient  ? 
WVion  thev  have  different  signs,  what  will  be  the  sign  of  the  quotient  ? 


DIVISION.  43 

3rd.  When  the  term  of  the  dividend  has  the  sign  —  ,  and 
ihat  of  the  divisor  the  sign  -}-,  the  quotient  must  have  the 
sign  —  .  Again  we  say  for  brevity,  that, 

+  divided  by  -}-,  and  —  divided  by  —  ,  give  +  ; 
—  divided  by  -f-,  and  -f-   divided  by  —  ,  give  —  . 

EXAMPLES. 

1.  Divide  4ax   by  —  2a.  Ans.  —  2a?. 
Here  it  is  plain  that  the  answer  must  be  —  2x  ;  for, 

—  2a  X  —  2x=  +  4ax,   the  divisor 

2.  Divide  3603z2   by  —  12a2*.  Ans.   —  3ax. 

3.  Divide  —  58a3b5c2d2   by  29a2£4c.  Ans.   —2abcd2. 

4.  Divide  —  84a4W3   by  —  42aWd.  Ans.  2aWdz. 

5.  Divide  64c*d*x3   by    16c*dx.  Ans.       4d*x2. 


6.  Divide    —  88£4*y   by    —  24&3c<fo5.      Ans.    +—f- 


7.  Divide  77a*y3z4   by    — ; 

8.  Divide  84a*b2c2d   by    —  42a4b2c2d. 

9.  Divide  —  60a72»<W   by  — : 

10.  Divide  —  88a667c6   by 

11.  Divide  16#2   by    —  8ar. 

12.  Divide  —  15a2#y3   by    Say. 

13.  Divide  —  84ab3x   by    —  I2b2. 

14.  Divide  —  96a4£2c3   by    I2a3bc. 

15.  Divide  —  144a968c7d5  by  —  36a4i6c6tZ. 

16.  Divide  256d3bc2x3   by    — 16a2ca:2.         Ans.    — \6abcx. 

17.  Divide  —  300a564c3*2  by  30aWc2x.      Ans.   —lOabcx. 
18  Divide  500a8/A-«  by   —  100a7i8c4.  Ans.   —5a?>c2. 


44  ELEMENTARY  ALGEBRA. 


19.  Divide   —  64a568c7  by  —  8a467c6.  Ans.  8abc. 

20.  Divide   -f-96«5W  by   —  24a*b2d.        Ans.   —  4</W. 

21.  Divide  72a5b3d*  by   —  8a*b2d.  Ans.   —  9abd3. 

Division  of  Polynomials. 

FIRST    EXAMPLE. 

47.  Divide     a2—  2ax+x2     by     a—x. 

It  is  found  most  convenient,  Dividend.     Divisor. 

in  division  in  algebra,  to  place         a2  —  2ax-}-x2  \a  —  x 


the  divisor  on  the  right  of  the         a2—  ax 

dividend,    and   the    quotient    di-  —   ax+x2  Quotient. 

rectly  u  ader  the  divisor.  —   ax-\-x2 


We  first  divide  the  term  a2  of  the  dividend  by  the  term  a 
of  the  divisor  :  the  partial  quotient  is  o,  which  we  place 
under  the  divisor.  We  then  multiply  the  divisor  by  a,  and 
subtract  the  product  a2  — ax  from  the  dividend,  and  to  the 
remainder  bring  down  x2.  We  then  divide  the  first  term  of 
the  remainder,  — ax  by  a,  the  quotient  is  — x.  We  then 
multiply  the  divisor  by  —a;,  and,  subtracting  as  before,  we 
find  nothing  remains.  Hence,  a  —  x  is  the  exact  quotient. 

In  this  example,  we  have  written  the  terms  of  the  dividend 
and  divisor  in  such  a  manner  that  the  exponents  of  the  same 
letter  shall  go  on  diminishing  from  left  to  right.  This  is 
what  is  called  arranging  the  dividend  and  divisor  with 
reference  to  a  certain  letter.  By  this  preparation,  the  first 
term  on  the  left  of  the  dividend,  and  the  first  on  the  left  of 
the  divisor^are  always  the  two  which  must  be  divided  bv 
each  other  in  order  to  obtain  a  term  of  the  quotient. 


QUEST. — 47.  What  do  you  understand  by  arranging  a  polynomial  with 
reference  to  a  particular  letter  ? 


DIVISION.  45 

48.  Hence,  for  the  division  of  polynomials  we  have  the 
following 

RULE. 

I.  Arrange  the  dividend  and  divisor  with  reference  to  a  cer- 
tain  letter,  and  then  divide  the  first  term  on  the  left  of  the 
dividend  by  the  first  term  on  the  left  of  the  divisor,  the  result 
is  the,  first  term  of  the  quotient ;   multiply  the  divisor  by  this 
term,  and  subtract  the  product  from  the  dividend. 

II.  Then  divide  the  first  term  of  the  remainder  by  the  first 
term  of  the  divisor,  which  gives  the  second  term  of  the  quotient ; 
multiply  the  divisor  by  the  second  term,  and  subtract  the  pro* 
duct  from  the  result  of  the  first  operation.      Continue  the  same 
process  until  you  obtain  0  for  a  remainder ;  in  which  case  the 
division  it  said  to  be  exact. 


SECOND    EXAMPLE. 

Let  it  be  required  to  divide 
51a2£2-}-1004  —  48a3b  —  15b*+4ab3    by    4ab— 
We  here  arrange  with  reference  to    a. 

Dividend.  Divisor. 

>  —  15M 


^   8a3b— 


— 2a?+8ab— 
Quotient. 


QUEST. — 48.  Give  the  general  rule  for  the  division  of  polynomials  1 
If  the  first  term  of  the  arranged  dividend  is  not  divisible  by  the  first  term 
of  *.he  arranged  divisor,  is  the  exact  division  possible  1  If  the  first  term 
of  any  partial  dividend  is  not  divisible  bv  the  first  term  of  the  divisor,  is 
the  e*act  division  possible  ? 

5* 


46  ELEMENTARY  ALGEBRA. 

REMARK. — When  the  first  term  of  the  arranged  dividend 
is  not  exactly  divisible  by  that  of  the*  arranged  divisor,  the 
complete  division  is  impossible  ;  that  is  to  say,  there  is  not 
a  polynomial  which,  multiplied  by  the  divisor,  will  produce 
the  dividend.  And  in  general,  we  shall  find  that  a  division 
is  impossible,  when  the  first  term  of  one  of  the  partial 
dividends  is  not  divisible  by  the  first  term  of  the  divisor. 

GENERAL   EXAMPLES. 

1.  Divide  I8xz   by    9x.  Ans.  2x. 

2.  Divide  Wx2y2   by    —  5x2y.  Ans.  —2y. 

3.  Divide  — 9ax2y2   by    9x2y.  Ans.  — ay. 

4.  Divide  —Sx2   by    —  2x.  Ans.  +  4x. 

5.  Divide  I0ab+I5ac   by    5a.  Ans.  2£  +  3c. 

6.  Divide  30aa?— 54x   by    6x.  Ans.  5a— 9. 

7.  Divide  I0x2y— I5y2— 5y  by  5y.  Ans.  2x2—3y  —  l. 

8.  Divide  \2a-{-3ax— I8ax2   by    3a.  Ans.  4  +  x— 6x2. 

9.  Divide  6ax2+ 9a2x+a2x2  by  ax.  Ans.  6x+9a  +  ax. 

10.  Divide    a2  -f  2 ax -\-x2   by    a-\-x.  Ans.  a+x. 

11.  Divide    a3  —  3a2y+3ay2— y3    by   a— y. 

Ans.  a2—2ay-{-y2. 

12.  Divide    24a2b  —  I2a3cb2— Gab   by    —  Gab. 

Ans.   —  4a+2a2c6-f  1. 

13.  Divide  6*4— 96  by  3*— 6.     Ans.  2ar3+4a:2+8x+16 

14.  Divide    .    .    .    a5— Sa^+lOaV— }0a2x3+5ax*—x* 
by     a2— 2ax+x2.  Ans.  a3  —  3a2x+3ax2—x3. 

15.  Divide     48x3  —  76ax2-64a2x+l05a3     by     2x-3a. 

Ans. 


DIVISION.  47 

16.  Divide  y6—  3y*x2+3y2x*—  *6  by  y3  —  3y2x+3yxz—  x*. 

Ans.  y3+3y 

17.  Divide  64aW—25a2bB   by   8/*2/»3+5<z64. 


18.  Divide    6a3+23a26+22a&2+553  by 

19.  Divide    6aff6+6a#2y6-f-42a2a?2    by 


20.  Divide    .    .  —  15a*+37a?bd—  29a2c/— 

Sc2/2   by    3a2—  5W+c/.  An^.   —  5a2+4W—  8c/. 

21.  Divide   o^  +  a&y+y*   by   a;2—  xy+y2. 

Ans.  x2  -\-xy-\-y2. 

22.  Divide    a?4—  y4   by    a:—  y.          Arcs.  a;3-f-rr2y-ha?y2-t-y3. 

23.  Divide    3a4-—  8a262-f  3a2c2+5i4--3i2c2   by    a2—  b2. 

Ans.  3a2—5b*+3c2. 

24.  Divide    .  .  6x6-5x5y2  —  6^4y4+6a;3y24-15a;3y3  —  9#2y4 
r10x2y5-f  15y6   by   Sa^-f  2a?2y2+3y2. 

Ans.  2x3—3x2y2+5y3. 

25.  Divide    .     —  c2+ldA2—  7abc—4a2bx—  6a2b2+6acx 
by   Sax  —  Sab  —  c.  Ans.  2ax-\-ab+c 

26.  Divide    ....    3x4-f-4a;3y—  4*2—  4a:2y2-j-16a:y-15 
by     2a:y+a;2—  3.  Ans.  3x2— 

27.  Divide     a;5+32y5     by     x+2y. 

Ans.  x*— 

28.  Divide     304—  26a3b  —  I4ab3+37a?b2     by     2b2—5ab    * 
4  3a2.  Ans.  a2—7ab. 


48  ELEMENTARY  ALGEBRA. 


CHAPTER  II. 

Algebraic  Fractions. 

49.  Algebraic  fractions  should  be  considered  in  the  same 
point  of  view  as  arithmetical  fractions,  such  as  J,  \±  ;  that 
is,  we  must  conceive  that  the  unit  has  been  divided  into  as 
many  equal  parts  as  there  are  units  in  the  denominator,  and 
that  one  of  these  parts  is  taken  as  many  times  as  there  are 
units  in  the  numerator.     Hence,  addition,  subtraction,  mul- 
tiplication, and  division,  are  performed  according  to  the  rules 
established  for  arithmetical  fractions. 

It  will  not,  therefore,  be  necessary  to  demonstrate  those 
rules,  and  in  their  application  we  must  follow  the  procedures 
indicated  for  the  operations  on  entire  algebraic  quantities. 

50.  Every  quantity  which  is    not  expressed  under  a 
fractional  form  is  called  an  entire  algebraic  quantity. 

51.  An  algebraic   expression,  composed  partly  of  an 
entire  quantity  and  partly  of  a  fraction,  is  called  a  mixed 
quantity. 


QUEST. — 49  How  are  algebraic  fractions  to  be  considered  1  What 
does  the  denominator  show  ?  What  does  the  numerator  show  ?  How 
then  are  the  operations  in  fractions  to  be  performed  1  50.  What  is  ars 
«ntire  quantity?  51.  What  is  a  mixed  quantity? 


ALGEBRAIC    FRACTIONS.  49 

CASE    I. 

To  reduce  a  fraction  to  its  simplest  terms. 

52.  A  fraction  is  said  to  be  in  its  lowest  terms,  when  there 
is  no  common  factor  in  the  numerator  and  denominator.  The 
rule  for  reducing  a  monomial  fractior  t  to  its  lowest  terms  has 
already  been  given  (Art.  44). 

With  respect  to  polynomial  fractions,  the  following  are 
cases  which  are  easily  reduced. 

1.  Take,  for  example,  the  expression 


This  fraction  can  take  the  form 

(a+b)  (a-b) 
(a-b)* 

(Art.  39  and  4O).     Suppressing  the  factor   a  —  b,  which 
is  common  to  the  two  terms,  we  obtain 

a+b 
a-b' 

2.  Again,  take  the  expression 


This  expression  can  be  decomposed  thus  : 


OT' 


Sa\a—b)        ' 

5a  (a-by 
8az(a-b)  ' 


QUEST. — 52.  How  do  you  reduced,  fraction  to  its  simplest  terms' 


50 


ELEMENTARY  ALGEBRA. 


Suppressing  the  common  factors  a(a  —  b),  the  result  is 

5(a-b) 

8a     ' 

Hence,  to  reduce  any  fraction  to  its  simplest  terms,  we  sup- 
press or  cancel  every  factor  common  to  the  numerator  and 
denominator. 

NOTE.  —  Find  the  common  factors  of  the  numerator  and 
denominator  as  explained  in  (Art.  41). 

EXAMPLES. 


1.  Keduce     -  —  -  —  „  ,  0      to  its  simplest  terms. 
12a4-{-6adc2 


2.  Reduce 


3.  Reduce 


4.  Reduce 


4a 
to  its  simplest  terms. 

An,. 


to  its  simplest  terms. 


to  its  simplest  terms. 


o.  Keauce 
7.  Reduce 


to  its  simplest  terms.     Ans.  —8. 
Ans.   


ALGEBRAIC  FRACTIONS.  51 

CASE  II. 

53.  To  reduce  a  mixed  quantity  to  the  form  of  a  fraction 

RULE. 

Multiply  the  entire  part  by  the  denominator  of  the  fraction  ; 
then  connect  this  product  with  the  terms  of  the  numerator  by 
the  rules  for  addition,  and  under  the  result  place  the  given 
denominator. 

EXAMPLES. 

1.  Reduce    6£   to  the  form  of  a  fraction 

A  Q 

6x7=42:     42  +  1=43:     hence,     6^=y. 

/fl2 X2\ 

2.  Reduce    x - —     to  the  form  of  a  fraction. 


XXX 

3.  Reduce-   x—  ^~—     to  the  form  of  a  fraction. 

2a 

ax-x* 

Ans.  . 

2<z 

4.  Reduce    5  -\ to  the  form  of  a  fraction. 

Ans.   — . 

5.  Reduce     1 to  the  form  of  a  fraction. 


Ans 

a 


QUEST. — 63.  How  do  you  reduce  a  mixed  quantity  to  the  form  of  a 
fraction  1 


2  ELEMENTARY  ALGEBRA. 

6.  Reduce     1-f  2x—  ^-     to  the  form  of  a  fraction. 

10*2+4ar-}-3 
Ans.   . 

7.  Reduce    2a-\-b —     to  the  form  of  a  fraction. 

16a+8b  —  3c— 4 
Ans.  _8 

8.  Reduce  6ax+ b to  the  form  of  a  fraction. 

40 

18azx+5ab 

Ans. 

4a 

9.  Reduce    8+3ab ° '     x       to  the  form  of  a  frac- 


tion, Ans.  , — 

I2abx4 

10    Reduce     9-\ — —     to  the  form  of  a  fraction. 

a—  b2 


CASE    III. 

54.  To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

RULE. 

Divide  the  numerator  by  the  denominator  for  the  entire  part, 
and  place  the  remainder,  if  any,  over  the  denominator  for  the 
fractional  part. 


QUEST. — 54.  How  do  you  reduce  a  fraction  to  an  entire  or  mixed 
quantity  1 


ALGEBRAIC  FRACTIONS.  53 

EXAMPLES. 


„    ,  8966 

1.  Keduce     — — —     to  an  entire  number. 

o 

8)8966( 


1120..  .6rem. 

Hence,  1120f=  Ans. 


2.  Reduce     to  a  mixed  quantity. 


Ans.  a . 


fig,  _  j, 

3.  Reduce     -     to  an  entire  or  mixed  quantity. 

Ans.  a 

ab—2cP 

4.  Reduce     -  -  -    to  a  mixed  quantity. 


2a2 

Ans.  a  --  =  —  . 
o 


_ 

5.  Reduce     -     to  an  entire  quantity.     Ans. 

3  _    3 

6.  Reduce     -  ?—    to  an  entire  quantity. 

x—  y 

Ans. 


10*2— 
7    Reduce     -  —  -     to  a  mixed  quantity. 


Ans.  2x—  H—  . 


8.  Reduce     —       — - —  to  a  mixed  quantity. 

Ans.  4x2—8-\ ^-. 


54  ELEMENTARY  ALGEBRA. 


t  CASE    IV. 

55.  To  reduce  fractions  having  different  denominators 
to  equivalent  fractions  having  a  common  denominator. 

RULE. 

Multiply  each  numerator  into  all  the  denominators  except  us 
own,  for  the  new  numerators,  and  all  the  denominators  togethe* 
for  a  common  denominator. 

EXAMPLES. 

1.  Reduce  ^,  £,  and  J,  to  a  common  denominator. 

1x3x5  =  15  the  new  numerator  of  the  1st. 
7x2x5=70  „  „  „  2nd. 

4x3x2=24  „  „  „  3rd. 

and      2x3x5=30  the  common  denominator. 

Therefore,  ^,  J$,  and  §£,  are  the  equivalent  fractions. 

NOTE.  —  It  is  plain  that  this  reduction  does  not  alter  the 
values  of  the  several  fractions,  since  the  numerator  and 
denominator  of  each  are  multiplied  by  the  same  number. 

2.  Reduce     ~     and    —    to  equivalent  fractions  having 
a  common  denominator. 

axc=ac    the  new  numerators. 


bxb=b* 
and  bxc=bc     the  common  denominator. 

QUEST.  —  55.   How  do  you  reduce  fractions  to  a  common  denominator  1 


ALGEBRAIC    FRACTIONS.  55 

Hence,     T—     and    —     are  the  equivalent  fractions. 
be  be 

3.  Reduce     —     and     to  fractions  having  a  com- 

ae ab+bz 

mon  denominator.  Ans.  —     and    — = . 

be  be 

4.  Reduce     — -,     — ,     and     rf,     to   fractions   having   a 

tliCL         oC 

9cx         4ab  .     6acd 

common  denominator.        Ans.  -- — ,     — — ,     and     — — . 

6ac         6ac  6ac 

5.  Reduce     — ,     — ,     and     a-\ ,     to  fractions  having 

43  a 

a  common  denominator. 

9a          8ax  I2a2 -\-24x 

Ans.  — — ,     -— — ,     and 


I2a  '      I2a  '  I2a 

1         a2  a2-\-x2 

6.  Reduce      — ,      -— ,      and ,       to    fractions 

having  a  common  denominator. 

3a+3x       2a3  +  2a2x  6a2+6a;2 

otz  ~r~  o^r       \ya  ~\~  \)oc  ocz  ~r~  \}oc 

a         6ax  a2—x2 

7.  Reduce     — p,     ,     and     = —     to   a   common 

30         5c  a 

denominator, 

5acd        ISabdx  \5a2bc—15bcx* 

AnS'  -^~^-»       15M  ' 


8.  Reduce     — ,     ,     and     — — r-,     to  a  common  de- 

5a          c  a-\-o 

nominator. 

5a3-5a&2 


56  ELEMENTARY  ALGEBRA. 

CASE    V. 

56.  To  add  fractional  quantities  together 

RULE. 

Reduce  the  fractions,  if  necessary,  to  a  common  denomina- 
tor;  then  add  the  numerators  together,  and  place  their  sum 
over  the  common  denominator. 

EXAMPLES. 

1.  Add  f,  f,  and  f  together. 
By  reducing  to  a  common  denominator,  we  have 

6x3x5  =  90  1st  numerator. 

4  X  2  x  5= 40  2nd  numerator. 

2x3x2  =  12  3rd  numerator. 

2x3x5  =  30  the  denominator. 

Hence,  the  fractions  become 

90     40     12       142 


30     30     30       30 
which,  by  reducing  to  the  lowest  terms  become  4|-|. 
2.  Find  the  sum  of    — ,     — ,     and     -~^ 

Here      axdxf=adf\ 

c  X  b  X  f=cbf  >  the  new  numerators 

exbxd=ebd  ) 
And        b  x  d  xf=  bdf     the  common  denominator. 

adf       cbf       ebd       adf+cbf+ebd     , 

=-  ~~ 


QUEST. — 56.  How  do  you  add  fractions. 


ALGEBRAIC    FRACTIONS.  67 


3    To     a-  add 


h  c, 

,  .  ,  2abx—3cx2 
Ans.  a  +  b-\  --  1  -  . 
be 

4.  Add     —  ,     —    and    -—     together.  Ans.  x-}-—. 

Z          o  4  \.<i 

x—2  4.u  I9x—  14 

5.  Add     —  -  —    and    —     together.  Ans.  —  —  -  » 

o  7  *1 

x—  2  2*—  3  ,  lOz-17 

6.  Add  a?H  --  to  3x-{  --  -  —  .         Ans.  4x-{  --  —  —  . 

34  I* 

7.  It  is  required  to  add   4#,  -  ,    and     -  together. 


2  ax 


,    ,    .  ,.  .  , 

8.  It  is  required  to  add     —  ,     —  ,  and    —  -  —   together. 

49a?-fl~2 

Ans. 


9.  It  is  required  to  add    4x,  —  ,  and  2+—   together. 

44^+90 
Ans. 


10.  It  is  required  to  add    3x-\  --  and  x  —  —   together! 

o  9 

Ans.  3x-\ 

4  O 

11.  Required  the  sum  of    ac  —  —  -     and     1  --  —. 

oa  a 

8a2cd—6bd+8ad—8ac 
Ans.   - 


Sad 


ELEMENTARY  ALGEBRA. 
CASE    VI. 

57,  To  subtract  one  fractional  quantity  from  another 

RULE. 

I.  Reduce  the  fractions  to  a  common  denominator. 

II.  Subtract  the  numerator  of  the  fraction  to  be  subtracted 
from  the  numerator  of  the  other  fraction,  and  place  the  dif- 
ference over  the  common  denominator. 


EXAMPLES. 

3  2 

1  .  What  is  the  difference  between     —  and  —  . 

3       2      24     14     10      5 

__M_LI  __  ___  _        n  ___  ^^.  ____     —  .  A  j\  « 

7       8  ~56     56~~56~28*  ' 

ff>  —  .  ft  2^2—  —  4-  /y* 

2.  Find  the  difference  of  the  fractions  —  —  and  -  . 

2b  3c 


(x—   a 

^  7  the  numerators 

(2a— 


And,  25x3c=z6Z>c      the  common  denominator. 

3cx—3ac      4ab—8bx     3cx—3ac—4ab  +  8bx 
Hence,  -  —  ---  -=•  -  =  --  -=•  --  .  Ans 
6bc  6bc  6bc 

,,,.,,.  f    I2x        ,   3o?  39a; 

3.  Required  the  difference  of   -  —  -  and  —  .     Ans.  —  —  . 

4.  Required  the  difference  of  5y  and   —  .      Ans.  —  ^-, 

8  8 

5.  Required  the  difference  of  —  and  —  .       Ans.  --  . 

79  63 


QUEST. — 57.  How  do  you  subtract  fractions  ? 


ALGEBRAIC    FRACTIONS.  69 

6.  Required  the  difference  between  —  -  —  and  —  =-. 

U  a 

dx-\-ad—bc 

Ans-  -  ~  —  • 


7.  Required  the  difference  of  —  —  —  and  —  -  —  . 

DO  o 

24x  f  8a—  Wbx—  355 
-          -406-        - 

J8.  Required  the  difference  of  3a?+—   and    x  --  . 

cx+  bx—  ab 
Ans.  2aH  --  7  --  . 

CASE    VII. 

58.  To  multiply  fractional  quantities  together. 

RULE. 

If  the  quantities  to  be  multiplied  are  mixed,  reduce  them  to 
fractional  forms  ;  then  multiply  the  numerators  together  for 
a  numerator  and  the  denominators  together  for  a  denominator 

EXAMPLES. 

1.  Multiply    i-    of    y    by     8J. 

Operation. 

We  first  reduce  the  com-  1.33 

pound  fraction  to  the  sim-  6             7  ~~~42' 

pie   one  ^,  and   then  the  25 

mixed  number  to  the  equiva-  8  J  =~3~" 
lent    fraction     2^  ;      after 

whicii,    we     multiply    the  Hence,     —  x—  =-^  =  ~. 

numerators  and  denomira-  42      3       126     42 

tors  together.  ^^^    2£ 


60  ELEMENTARY  ALGEBRA. 

2.  Multiply     «+*i    by    -L.     First,     a4^=?!d± 

a  d  a  a 

TT  a2-}-&»      c       a2c+bcx 

Hence,    .  — ' X-^= -, — .  Ans. 

a  d  ad 

.    3x  3a  9aa? 

3.  Kequired  the  product  of    —    and    -r-.        Ans.         •-. 

5  b  5b 

2  V  ^Y^ 

4.  Required  the  product  of   —    and    

o  2/<i 

3x* 

Ans.  — • — . 
5a 

.    2a?       Sab  Sac 

5.  Find  the  continued  product  of    — ,     and    — ~. 

a          c  2b 

Ans.  9ax. 

6.  It  is  required  to  find  the  product  of    b-\ and    — . 

a  x 

• 

ab+bx 
Ans. . 

x 

7.  Required  the  product  of    — ; and    —s . 

be  b-\-c 

** 
Ans. 


x  I  j  x  _  j 

8.  Required  the  product  of    x-\  --  ,    and  ,  . 

ax2—  ax+x*—  1 


9.  Required  the  product  of    a-\  --    by 


a-x  p+aT 

n.    — —  n-T* 

A™. 


— x2 — x3 


QUEST  — 6S.  How  do  yoa  multiply  fractions  together? 


ALGEBRAIC    FRACTIONS.  61 

CASE    VIII. 

59.  To  divide  one  fractional  quantity  by  another. 

RULE.  , 

Reduce  the  mixed  quantities,  if  there  are  any,  to  fractional 
forms  ;  then  invert  the  terms  of  the  divisor,  and  multiply  the 
fractions  together  as  in  the  last  case, 

EXAMPLES. 

10  5 

1.  Divide.    .     .     .     —   by    —  . 

If  the  divisor  were  5,  the  Operation. 

quotient  would  be  y1^.     But,  5  _         1 

since  the  divisor  is  £  of  5,  8  ~          8 

the  true  quotient  must   be  8  10  10 

times  11^,  for  the  eighth  of  24  ~     ~~12(J 

a  number  will  be  contained  10  80        2 

^       __      \^  O  _  __     _  -—  —  ___ 

'\n  the  dividend  8  times  more  120          "120       3  ' 

than  the  number  itself.     In 

this  operation  we  have  actually  multiplied  the  numerator 

of  the  dividend  by  8  and  the  denominator  by  5  ;  that  is,  we 

have  inverted  the  terms  of  the  divisor  and  multiplied  the  fractions 

together. 

2.  Divide     .     .     a—--   by    —  . 

2c  g 

b      2ac—b 


b        f      2ac  —  b 
Hence,     ±-+±=-— 


QUEST. — 59.  How  do  you  divide  one  fraction  by  another  1 


62  ELEMENTARY  ALGEBRA. 

7x                               12  91a? 

3.  Let     -—     be  divided  by    — .  Ans    -^--. 

5                                  13  60 

4.  Let     — - —     be  divided  by     5a?.  Ans.  —-. 


5.  Let     —TT-     be  divided  by    — .  Ans.  — — 

6.  Let be  divided  by     — -.  Ans.  -, 

o?—l  2  a;— 1 

5x  2a  5bx 

7.  Let     —     be  divided  by     -r-.  Ans.  -— — 


-=-. 
60 


x  —  b  3cx  x  —  b 

8.  Let     -  -     be  divided  by     —  j-.  Ans. 

8cd  4d 


9.  Let      2      -  be  divided  by  . 

x2— 2bx-\-  W-  x—b 

Ans.  x-\ . 

10.  Divide      6<z2  +  —     by     c2— ^^. 
5  2 


11.  Divide      18c2—a:  +  -^-     by     a2 . 


12.  Divide      20*2  —  -j^-     by     a?2 -^. 

Ant     J£ 


EQUATIONS    OF    THE    FIRST    DEGREE.  63 


CHAPTER  III. 


Of  Equations  of  the  First  Degree. 

GO.  An  Equation  is  the  algebraic  expression  of  two  equal 
quantities  with  the  sign  of  equality  placed  between  them. 
Thus,  x=a+b  is  an  equation,  in  which  x  is  equal  to  the 
sum  of  a  and  b. 

6 1 .  By  the  definition,  every  equation  is  composed  of  two 
parts,  separated  from  each  other  by  the  sign   =.     The  part 
on  the  left  of  the  sign,  is  called  the  first  member ;  and  the 
part  on  the  right,  is  called  the  second  member.     Each  mem- 
ber may  be 'composed  of  one  or  more  terms.     Thus,  in  the 
equation   x=a-\-b,  x   is  the  first  member,  and   a-\-b   the 
second. 

62.  Every  equation  may  be  regarded  as  the  enunciation, 
in  algebraic  language,  of  a  particular  question.     Thus,  the 
equation    a: +#  =  30,    is  the    algebraic    enunciation  of  the 
following  question : 

QUEST. — 60.  What  is  an  equation1!  61.  Of  how  many  parts  is  every 
equation  composed  1  How  are  the  parts  separated  from  each  other  1 
What  is  the  part  on  the  left  called  1  What  is  the  part  on  the  right  called  ? 
May  each  member  be  composed  of  one  or  more  terms  1  In  the  equation 
x=a-\-b,  which  is  the  first  member?  Which  the  second?  How  many 
terms  in  the  first  member?  How  many  in  the  second1 


64  ELEMENTARY  ALGEBRA 

To  find  a  number  which  being  added  to  itself,  shall  give,  a 
sum  equal  to  30. 

Were  it  required  to  solve  this  question,  we  should  first 
express  it  in  algebraic  language,  which  would  give  the 
equation 

x+x=30. 

By  adding  x  to  itself,  we  have 
2* =30. 

And  by  dividing  by  2,  we  obtain 
cc=\.5. 

Hence  we  see  that  the  solution  of  a  question  by  algebra 
consists  of  twa  distinct  parts :  viz.  the  STATEMENT  and  the 
SOLUTION  of  an  equation. 

I.  The  STATEMENT  consists  in  expressing  algebraically 
the  relation  between  the  knoivn  and  unknown  quantities. 

II.  The  SOLUTION  of  the  equation  consists  in  finding  the 
value  of  the  unknown  quantity  in  terms  of  those  which  are 
known. 

The  given  or  known  parts  of  a  question,  are  represented 
either  by  numbers  or  by  the  first  letters  of  the  alphabet, 
a,  b,  c,  &c.  The  unknown  or  required  parts  are  repre- 
sented by  the  final  letters,  x,  y,  z,  &c. 

EXAMPLE. 

Find  a  number  which,  being  added  to  twice  itself,  the 
sum  shall  be  equal  to  24. 


QUEST. — 62.  How  may  you  regard  every  equation  ?  What  question 
does  the  equation  x+x=3Q  state"!  Of  how  rr.any  parts  does  the  solu- 
tion of  a  question  by  algebra  consist  1  Name  them.  What  is  the  2nd 
part  called  1  By  what  are  the  known  parts  of  a  question  represented  ? 
By  what  are  the  unknown  parts  represented  ? 


EQUATIONS    OF    THE    FIRST    DEGREE.  05 

Statement. 
Let  x  represent  the  number.     We  shall  then  have 


This  is  the  statement. 

Solution. 

Having    ....     x+2x=24, 
we  add    .....     a;+2a?, 
which  gives      .     .     .  3^=24  , 

and  dividing  by  3,      .  x—8. 

63.  An  equation  is  said  to  be  verified  when  the  answer 
found,  being  substituted  for  the  unknown  quantity,  proves 
the  two  members  of  the  equation  to  be  equal  to  each  other. 

Thus,  in  the  last  equation  we  found  #=8.  If  we  substi- 
tute this  value  for  x  in  the  equation 


we  shall  have  8  +  2x8  =  8  +  13  =  24. 

which  proves  that  8  is  the  true  answer. 

6  4.  An  equation  involving  only  the  first  power  of  the 
unknown  quantity,  is  called  an  equation  of  the  first  degree. 

Thus,  6a;+3a?—  5  =  13, 

and  ax-}-  bx  +  c  =  d, 

are  equations  of  the  first  degree. 

By  considering  the  nature  of  an  equation,  we  perceive 
that  it  must  possess  the  three  following  properties  : 

QUEST.  —  63.  When  is  an  equation  said  to  be  verified  1  64.  When 
an  equation  involves  only  the  first  power  of  the  unknown  quantity,  wnat 
is  it  called  ?  What  are  the  three  properties  of  every  equation  ? 

7 


66  ELEMENTARY  ALGEBRA. 

1  st.  The  two  members  are  composed  of  quantities  of  the 
same  kind  :  that  is,  dollars  =  dollars,  pounds  =  pounds,  &c. 
2nd.  The  two  members  are  equal  to  each  other. 
3rd.  The  two  members  must  have  the  same  sign. 

65.  An  axiom  is  a  self-evident  truth.     We  may  here 
state  the  following. 

1.  If  equal  quantities  be  added  to  loth  members  of  an  equa- 
tion, the  equality  of  the  members  will  not  be  destroyed. 

2.  If  equal  quantities  be  subtracted  from  both  members  of 
an  equation,  the  equality  will  not  be  destroyed. 

3.  If  both  members  of  an  equation  be  multiplied  by  the  same 
number,  the  equality  will  not  be  destroyed. 

4.  If  both  members  of  an  equation  be  divided  by  the  same 
number,  the  equality  will  not  be  destroyed. 

Transformation  of  Equations. 

66.  The  transformation  of  an  equation  consists  in  chang 
ing  its  form  without  affecting  the  equality  of  its  members. 

The  following  transformations  are  of  continual  use  in  the 
resolution  of  equations. 

First  Transformation. 

67.  When  some  of  the  terms  of  an  equation  are  frac- 
tional, to  reduce  the  equation  to  one  in  which  the  terms  shall 
be  entire. 

1.  Take  the  equation 


3 


QUEST.  —  65.  What  is  an  axiom  ?  Name  the  four  axioms.  66.  Whaf 
is  the  transformation  of  an  equation  ?  67.  What  is  the  first  transforma- 
tion ?  What  is  the  least  common  multiple  of  several  numbers  1  How 
do  von  find  the  least  common  multiple  7 


EQUATIONS    OF    THE    FIRST    DEGREE.  67 

First,  reduce  all  the  fractions  to  the  same  denominator 
by  the  known  rule  ;  the  equation  then  becomes 

48s        54a?    ,    12a;__11 
"72          72         72"- 

and  sinc«  we  can  multiply  both  members  by  the  same  num- 
ber without  destroying  the  equality,  we  will  multiply  them 
by  72,  which  is  the  same  as  suppressing  the  denominator 
72,  in  the  fractional  terms,  and  multiplying  the  entire  term 
by  72  ;  the  equation  then  becomes 

48a?—  54a?+12a?=792, 
or  dividing  by  6       Sx—  9a?-f-  2#=132. 

But  this  last  equation  can  be  obtained  in  a  shorter  way,  by 
finding  the  least  common  multiple  of  the  denominators. 

The  least  common  multiple  of  several  numbers  is  the 
least  number  which  they  will  separately  divide  without  a 
remainder.  When  the  numbers  are  small,  it  may  at  once 
be  determined  by  inspection.  The  manner  of  finding  the 
least  common  multiple  is  fully  shown  in  Arithmetic  §  87 

Take  for  example,  the  last  equation 

2x      3         x 
---  #-!_  -—=11. 
346 

We  see  that  12  is  the  least  common  multiple  of  the  deno- 
minators, and  if  we  multiply  all  the  terms  of  the  equation 
by  12,  and  divide  by  the  denominators,  we  obtain 


8a;—  9. 
the  same  equation  as  before  found. 


08  ELEMENTARY    ALGEBRA. 

68.  Hence,  to  make  the  denominators  disappear  from  an 
equation,  we  have  the  following 


RULE. 


I.  Find  the  least  common  multiple  of  all  the  denomi' 
nators. 

II.  Multiply  every  term  of  the  equation  by  the  common 
multiple  —  reducing  at  the  same  time  the  fractional  to  entire 
terms. 

EXAMPLES. 

1.  Clear  the  equation  of    -—  —  f—  —  4  =  3     of  its  denomi- 

o        i. 

nators.  Ans.  7x+5x—  140^105. 

2.  Clear  the  equation     -^-+-r  —  ^;=8     °f  its   denomi- 

o      y     4ti 

nators.  Ans.  9x  -\-6x-  2x  =  432 

3.  Clear  the  equation     --  f—  ---  jrH  —  r^20     ofitsde- 

-w  O  3  1  <t 

nominators.  Ans. 


4.  Clear  the  equation     —  —  f  —  —  —  =4     of  its   denomi- 

O         7         Z 

nators.  Ans.   14x+lQx—  35ar=280. 

5.  Clear  the  equation     —  ---  -  —  ^--—=zl5     of  its  denomi- 

4       5       o 

nators  Ans.  I5x  — 


QUEST. — 68.  Give  the  rule  for  clearing  an  equation  of  its  denoini 
nators. 


EQUATIONS    OF    THE    FIRST    DEGREE.  69 

6.  Clear  the  equation     -^ TT+'S'+'Tr  — 12     of  its  de~ 

4      o      o      y 

nominators.  Ans.  18o?— 12*4-  9oj-f-8a:=864 

7.  Clear  the  equation     — —+f=g. 

Ans.  ad—bc+bdf=bdg. 

8.  In  the  equation 


ax      2c2x  4bc*x      5a3    ,    2c2 

7 — K4a  =  — 5 rr — ! 3*» 

b         ab  a3  b2  a 

the  least  common  multiple  of  the  denominators  is   a3b* ; 
hence  clearing  the  fractions,  we  obtain 

= 4b3c2x— 


Second  Transformation. 

69.  When  the  two  members  of  an  equation  are  entire 
polynomials,  to  transpose  certain  terms  from  one  member  to 
the  other. 

1.  Take  for  example  the  equation 
5x— 6  =  8  +  2*. 

If,  in  the  first  place  we  subtract  2x  from  both  members, 
the  equality  will  not  be  destroyed,  and  we  have 

5x— 6— 2x^8. 

Whence  we  see  that  the  term  2ar,  which  was  additive 
in  the  second  member  becomes  subtractive  in  the  first. 


QUEST. — 69.  What  is  the  second  transformation?      What  do   you 
understand  by  transposing  a  term  1     Give  the  rule  for  transposing  from 

one  member  to  the  other. 

7* 


70  ELEMENTARY  ALGEBRA. 

In  the  second  place,  if  we  add  6  to  both  members, 
equality  will  still  exist,    and  we  have 

5x— 6  — 2^+6  =  8  +  6. 

Or,  since   — 6   and  -{-6   destroy  each  other,  we  have 
5x— 2x= 

Hence  the  term  which  was  subtractive  in  the  first  mem- 
ber, passes  into  the  second  member  with  the  sign  of 
addition. 

2.  Again,  take  the  equation 

ax-\-l  =  d — ex. 

If  we  add  ex  to  both  members  and  subtract  b  from 
them,  the  equation  becomes 

ax-\-b-\-cx — b=zd — cx-\-cx — b. 
or  reducing  ax-}-cx=:d — b. 

When  a  term  is  taken  from  one  member  of  an  equation 
and  placed  in  the  other,  it  is  said  to  be  transposed. 

Therefore,  for  the  transposition  of  the  terms,  we  have  the 
following 

RULE. 

Any  term  of  an  equation  may  be  transposed  from  one  mem- 
ber to  the  other  by  changing  its  sign. 

7O.  We  will  now  apply  the  preceding  principles  to  the 
resolution  of  equations. 

1.  Take  the  equation 


EQUATIONS    OF    THE    FIRST    DEGREE.  71 

By  transposing  the  terms    —  3    and   2#,  it  becomes 

4x—  2x=5  +  3. 
Or,  reducing  2x  =  8. 

0 

Dividing  by  2  x  =—=4. 

Verification. 

If  now,   4  be  substituted  in  the  place  of  x  in  the  given 
equation 


it  becomes  4x4—3=2x 

or,  13  =  13. 

Hence,  the  value  of  x   is  verified  by  substituting  it  for  the 
unknown  quantity  in  the  given  equation. 

2.  For  a  second  example,  take  the  equation 

5x     4x  7        I3x 

12      3  "        ~  8  ~     6    ' 

By  making  the  denominators  disappear,  we  have 

10*—  32*—  312=21—  52*, 
or,  by  transposing 

Wx—  32a?+52#=:21  +  312 
by  reducing  30a?  ==  333 

333       111 
~-30-~-~-lO-~-   U>1' 

a  result  which  may  be  verified  by  substituting  it  for  x  in  the 
given  equation. 

3.  For  a  third  example  let  us  take  the  equation 

(3a—  x)(c  — 


ELEMENTARY  ALGEBRA. 

It  is  first  necessary  to  perform  the  multiplications  indica- 
ted, in  order  to  reduce  the  two  members  to  two  polynomials, 
and  thus  be  able  to  disengage  the  unknown  quantity  x,  from 
the  known  quantities.  Having  done  that,  the  equation 
becomes, 

3a2—  ax— 
or,  by  transposing 


by  reducing  ax  —  3bx  =  7ab  —  3a*. 

Or,  (Art.  41).  (a—  3b)x  =  7ab  —  3az. 

Dividing  both  members  by    a  —  3b   we  find 

_7ab—  3a? 
x~     a—36   ' 

Hence,  in  order  to  resolve  an  equation  of  the  first  degree, 
we  have  the  following  general 

RULE. 

I.  If  there  are  any  denominators,  cause  them  to  disappear, 
and  perform,  in  both  members,  all  the  algebraic  operations 
indicated. 

II.  Then  transpose  all  the  terms  affected  with  the  unknown 
quantity  into  the  first  member,  and  all  the  known  terms  into 
the  second  member. 

III.  Reduce  to  a  single  term  all  the  terms  involving  x  : 
this  term  will  be  composed  of  two  factors,  one  of  which  will 
be  x,  and  the  other  all  the  multipliers  of  x,  connected  with 
their  respective  signs. 

IV.  Divide  both  members  of  the  equation  by  the  multi* 
plier  of  the  unknown  quantity. 

QUEST.  —  70.  What  is  the  first  step  in  resolving  an  equation  of  the 
first  degree  ?    What  the  second  ?    What  the  third  ?    What  the  fourth  ? 


EQUATIONS  OF  THE  FIRST  DEGREE.        73 
EXAMPLES. 

1.  Given   3x— 2+24=31    to  find   a:.  Ans.  x=3. 

2.  Given   a?+18  =  3a:— 5    to  find   x.          Ans.  a?=lly. 

3.  Given    6— 2ar+10=20— 3x— 2    to  find   x. 

Ans.  x=2. 


6.  Given     2x x-\-L=5x — 2  to  find  x.     Ans.  #=-=-. 

2  7 

6.  Given     3ax-\ 3  =  for— a     to  find     x. 

6-3a 
Ans.  x  =  - 


7.  Given       z+-=20— z          to  find 
23  2 


8.  Given  4.=4-  to  find 

£          3  4 


9.  Given     - — --|-ar=~-- 3     to  find     x. 
4        « 


10.  Given     — ^ ^ — 4=/    to  find    a:. 


6a—2l>' 


cdf  \-4cd 

Ans.  x=     J  -. 

3ad—2bc 


74  ELEMENTARY  ALGEttRA. 


n.  Given     !2^±--2*rL=4-*     to  find     *. 


56  +  95— 7c 
:         16a        ' 


12.  Given    -^ — -~5 — l"7r=-5-     to  find     x. 
O          3          2        3 


An*.  a;  =10. 

13.  Given    -  —  4~+~  —  ^T=f    to  find     *• 

a       be       d     J 

__  abcdf 

~  bcd-acd+abd-abc* 

NOTE.  —  What  is  the  numerical  value  of  x,  when  a=l, 
5=2,  c=3,  d=4,  5=5,  and/=6. 

14.  Given    ^-^-^i=-12ff     to  find    *. 

7      y       *> 

.  «=14. 


15.  Given     a-+^+l     to  find     «. 


16.  Given     ar+i+i—  ^-=2^-43     to  find     *. 

4       5       o 

An*,  a:  =60 

17.  Given     2a?  --  ^~     =  —  -  —     to  find     x. 

5  £ 


Ans.  x=3. 


18    Given     3x-\ — ~ =x+a     to  find     x. 

o 


ax — b       a      bx       bx — a  -.    , 

19.  Given     — ^=~2 3 —    **  *' 

35 
Ans.  x=- 


3a  -! 


EQUATIONS    OF    THE    FIRST    DEGREE.  75 

20.  Find  the  value  of  x  in  the  equation 

(a+b)  (x—b)  4ab—b2  az—bx 

-  -  --  3a  =  -  •-—  --  2x-\  --  -  -  . 
a  —  b  a-\-o  b 


Of  Questions  producing  Equations  of  the  First  Degree 
involving  but  a  single  unknown  quantity. 

71.  It  has  already  been  observed  (Art.  62),  that  the 
solution  of  a  question  by  algebra  consists  of  two  distinct 
parts  : 

1st.  To  make  the  STATEMENT  :  that  is,  to  express  the  con- 
ditions of  the  question  algebraically  ; 

2d.  To  solve  the  equation  :  that  is,  to  disengage  the  known 
from  tne  unknown  quantities. 

We  have  already  explained  the  manner  of  finding  the 
value  of  the  unknown  quantity,  after  the  question  has  been 
stated  ;  and  it  only  remains  to  point  out  the  best  methods 
of  putting  a  question  in  the  language  of  algebra. 

This  part  of  the  algebraic  solution  of  a  question  cannot, 
like  the  second,  be  subjected  to  any  well  defined  rule. 
Sometimes  the  enuroiation  of  the  question  furnishes  the 
equation  immediately  ;  and  sometimes  it  is  necessary  to  dis- 
cover, from  the  enunciation,  new  conditions  from  which  an 
equation  may  be  formed. 

QUEST.  —  71.  Into  how  many  parts  is  the  resolution  of  a  question  in 
a.gebra  divided  1  What  is  the  first  step  ?  Whai  the  second  1  Which 
part  has  already  been  explained  ?  Which  part  is  now  to  be  considered  ? 
Can  this  part  be  subjected  to  exact  rules  ?  Give  the  general  rule  for 
stating  a  question 


76  ELEMENTARY  ALGEBRA. 

In  almost  all  cases,  however,  we  are  able  to  make  the  state- 
ment ;  that  is,  to  discover  the  equation,  by  applying  the  fo/- 
lowing 

RULE. 

Represent  the  unknown  quantity  by  one  of  the  final  letters 
of  the  alphabet ;  and  then  indicate  by  means  of  the  algebraic 
signs,  the  same  operations  on  the  known  and  unknown  quan- 
tities, as  would  verify  the  value  of  the  unknown  quantity, 
were  such  value  known. 

QUESTIONS. 

1.  To  find  a  number  to  which  if  5  be  added,  the  sum  will 
be  equal  to  9. 

Denote  the  number  by     x. 
Then  by  the  conditions 


This  is  the  statement  of  the  question. 
To  find  the  value  of  x,  we  transpose  5  to  the  second 
member,  which  gives 

a;  =  9-5  =  4. 

Verification. 
4+5  =  9. 

2.  Find  a  number  such,  that  the  sum  of  one  half,  one 
third,  and  one  fourth  of  it,  augmented  by  45,  shall  be  equal 
to  448. 

Let  the  required  number  be  denoted  by     x. 

Then,  one  half  of  it  will  be  denoted  by     — -. 

one  third  „  „  by    -—. 

o 

one  fourth  by     — -. 


EQUATIONS    OF    THE    FIRST    DEGREE  77 

And  by  the  conditions. 


This  is  the  statement  of  the  question. 
To  find  the  value  of  x,  subtract  45  from  both  members  : 
this  gives 


By  clearing  the  terms  of  their  denominators,  we  obtain 

6a:+4a;+3a:=4836, 
or  13a:=4836. 

Hence,  x=  —  —=372. 

Verification. 

372        372        372 

—+-—+-±-1+45  =  186  +  124+93  +  45=448. 

4i  O  4 

3.  What  number  is  that  whose  third  part  exceeds  its 
fourth  by  16. 

Let  the  required  number  be  represented  by  x.     Then, 

-—#=     the  third  part. 
3 

1 
-j-a?=     the  fourth  part. 

And  by  the  question 


This  is  the  statement.     To  find  the  value  of  a?,  we  clear 
the  terms  of  the  denominators,  which  gives 

4a?—  3a;=192. 
and  a:  =192. 

8 


78  ELEMENTARY  ALGEBRA. 

Verification. 


4.  Divide  $1000  between  A,  B   and  C,  so  that  A  shall 
have  $72  more  than  B,  and  C  $100  more  than  A. 

Let  x  =  B's  share  of  the  $1000. 

Then  x+  72=     A's  share, 

and  a:+172  =     C's  share, 

their  sum  is          3a?+244=$1000. 
This  is  the  statement. 
By  transposing  244  we  have 
3x=  1000—  244=756 

py  f  /> 

and          x  =-——=252=  B's  share. 

9 

Hence,    a+    72  =252+   72  =  $324=     A's  share. 
And         *-t-  172  =252-{-172  =  $424=     C's  share. 

Verification. 
252  +  324+424  =  1000. 

5.  Out  of  a  cask  of  wine  which  had  leaked  away  a  third 
part,  21  gallons  were  afterwards  drawn,  and  the  cask  being 
then  guaged,  appeared  to  be  half  full:  how  much  did  it 
hold? 

Suppose  the  cask  to  have  held     x     gallons. 

••Then,         —  -    what  leaked  away. 

0 

x 

And  ~+  21=     what  had  leaked  and  been  drawn 

3 

Hence,       •—+  21=—  -x    by  the  question. 
o  2 

This  is  the  statement. 


EQUATIONS    OF    THE    FIRST    DEGREE.  79 

To  find  #,  we  have 

2.c+126=:3tr, 

and  —  x  =—126, 

or  x  =     126, 

by  changing  the  signs  of  both  members,  which  does  not 
destroy  their  equality. 

Verification. 


6.  A  fish  was  caught  whose  tail  weighed  9lb.,  his  head 
weighed  as  much  as  his  tail  and  half  his  body,  and  his  body 
weighed  as  much  as  his  head  and  tail  together  ;  what  was 
the  weight  of  the  fish  ? 

Let  2ar=    the  weight  of  the  body. 

Then,  9-{-x=   weight  of  the  head  ; 

and  since  the  body  weighed  as  much  as  both  head  and  tail, 


which  is  the  statement.     Then, 

2x—  x=lS     and     ar=18. 

Verification. 

2x  =  36/6.  =  weight  of  the  body. 

9-f  ar=27/6.=  weight  of  the  head. 

9lb.=  weight  of  the  tail. 

Hence,  72/6.  =  weight  of  the  fish. 


80  ELEMENTARY  ALGEBRA. 

7.  The  sum  of  two  numbers  is  67  and  their  difference 
19  :  what  are  the  two  numbers  ? 

Let  x=    the  least  number. 

Then,  #-f-19  =    the  greater, 

and  by  the  conditions  of  the  question 

2*+ 19  =  67. 
This  is  the  statement. 

To  find  a?,  we  first  transpose  19,  which  gires. 
2a?=67  — 19=48; 

4R 

hence,         *=-— =  24,     and     x+t9=43. 

Verification. 
434-24=67,     and     43—24=19. 

Another  Solution. 

Let   a?   represent  the  greater  number  : 
then,  or— 19   will  represent  the  least, 

and,  2a?— 19=67,   whence   2;r=67-f-I9; 

fifi 

therefore,  a:= — =43 

2 

and  consequently     x — 19=43  —  19=24. 

General  Solution  of  this  Problem. 

The  sum  of  two  numbers  is    «,   their  difference  is    b. 
What  are  the  two  numbers  ? 


EQUATIONS    OF    THE    FIRST    DEGREE.  81 

Let   x   be  the  least  number, 

x-\-b   will  represent  the  greater. 
Hence,          2x  +  h=a,   whence   1x—a  —  &; 


a  —  b       a       b 
therefore,  a=  —=_--, 

a       b  a       b 

and  consequently,    x-\-b=  ---  —  +6:=—  -}-—-. 
2        2  2       « 

As  the  form  of  these  two  results  is  independent  of  any 
value  attributed  to  the  letters  a  and  b,  it  follows  that, 

Knowing  the  sum  and  difference  of  two  numbers,  we  obtain 
the  greater  by  adding  the  half  difference  to  the  half  sum,  and 
the  less,  by  subtracting  the  half  difference  from  half  the  sum. 

Thus,  if  the  given  sum  were  237,  and  the  difference  99, 

237      99  237  +  99       336 

the  greater  is     -T+T'    or   -  -  r^-—  =  168; 


237      99  138 

and  the  least     —  ---  ,    or  =69. 

22  2 


Verification. 

168+69=237     and     168—69=99. 

8.  A  person  engaged  a  workman  for  48  days.  For 
each  day  that  he  laboured  he  received  24  cents,  and  for 
each  day  that  he  was  idle,  he  paid  12  cents  for  his  board. 
At  the  end  of  the  48  days,  the  account  was  settled,  when 
the  labourer  received  504  cents.  Required  the  number  of 
working  days,  and  the  number  of  days  he  was  idle. 
8* 


82  ELEMENTARY  ALGEBRA. 

If  these  two  numbers  were  known,  by  multiplying  them 
respectively  by  24  and  12,  then  subtracting  the  last  product 
from  the  first,  the  result  would  be  504.  Let  us  indicate 
these  operations  by  means  of  algebraic  signs.. 

Let  x  =    the  number  of  working  days. 

48  —  x  =    the  number  of  idle  days. 
Then,        24  x  oc  =    the  amount  earned, 
and       12(48  —  x)=    the  amount  paid  for  his  board. 
Then,       24#  —  12(48—  #)  =  504 

what  he  received,  which  is  the  statement.     Then  to  find  % 
we  first  multiply  by  12,  which  gives 

24a?—  576  +  I2a?=504. 


or,  36^=5044-576  =  1080,  . 

a?=—  —=30   the  working  days. 

oO 

whence,  48—30  =  18   the  idle  days. 

Verification. 

Thirty  day's  labour,  at  24  cents 
a  day,  amounts  to     ......     30x24=720  cents 

And  18  day's  board,  at  12  cents 
a  day,  amounts  to     ......     18  X  12=216  cents. 

The  difference  is  the  amount  received  504  cents. 


General  Solution. 

This  question  may  be  made  general,  by  denoting  the 
whole  number  of  working  and  idle  days  by    n. 

The  amount  received  for  each  day  he  worked  by  a. 
The  amount  paid  for  his  board,  for  each  idle  day,  by   b 


EQUATIONS    OF    THE    FIRST    DEGREE.  83 

And  the  balance  due  the  laborer,  or  the  result  of  the 
account,  by   c, 

As  before,  let  the  number  of  working  days  be  repre- 
sented by   a:. 

The  number  of  idle  days  will  be  expressed  by   n — x. 
.  Hence,  what  he  earns  will  be  expressed  by   ax. 

And  the  sum  to  be  deducted,  on  account  of  his  board,  by 

The  equation  of  the  problem  therefore  is 
ax — b(n  —  x)=c, 

which  is  the  statement.     To  find  x  we  first  multiply  by  6, 
which  gives 

ax — bn-\-bx=c 
or,  (a-\-b)x=c  -{-bn 

c   -\-bn  .  . 

whence,  x= — TTT~~~     working  days. 

c    -4-bn       an-4-bn — c^-bn 

and  consequently,  n— x=n -7—= — , 

<  a   -4-0  a+b 


an—c  ...     . 

or  n — x= —  =     idle  days. 

a-{-b 


Let  us  now  suppose  n  =  48,  a  =  24,  5  =  12,  and  c=r504 
These  numbers  will  give  for  x  the  same  value  as  before 
found. 

9.  A  person  dying  leaves  half  of  his  property  to  his  wite, 
one-sixth  to  each  of  two  daughters,  one-twelfth  to  a  servant 
and  the  remaining  $600  to  the  poor :  what  was  the  amount 
of  his  property? 


84  ELEMENTARY  ALGEBRA. 

Represent  the  amount  of  the  property  by  x. 

Then,         -— =         what  he  left  to  his  wife, 
Z 

-— -=         what  he  left  to  one  daughter, 
and  — =— -     what  he  left  to  both  daughters, 

D  o 

— =         what  he  left  to  his  servant. 

$600         to  the  poor. 
Then,  by  the  conditions  of  the  question 

Y+T+TI+600=* 

the  amount  of  the  property,  which  gives     x =$7200. 

10.  A  and  B  play  together  at  cards.  A  sits  down  with 
$84  and  B  with  $48.  Each  loses  and  wins  in  turn,  when 
it  appears  that  A  has  five  times  as  much  as  B.  How  much 
did  A  win  ? 

Let  x  represent  what  A  won. 
Then  A     rose  with     $84+ a?     dollars, 

and  B     rose  with     $48— a:     dollars. 

But  by  the  conditions  of  the  question,  we  have 

84+;r=5(48  — a:), 

hence,  84-j-o?=240  —  5#; 

and,  6*  =156, 

consequently,  oc  =  $26     what  A  won. 

Verification. 

84+26  =  110;     48-26=22; 
110  =  5(22)  =  110 


EQUATIONS    OF    THE    FIRST    DEGREE.  85 

11.  A  can  do  a  piece  of  work  alone  in  10  days,  B  in  13 
days  :  in  what  time  can  they  do  it  if  they  work  together  ? 
Denote  the  time  by  cc,  arid  the  work  to  be  done  by  1. 

Then  in  1  day  A  could  do     —     of  the  work,  and  B  could 
do     —  ;  and  in  x  days  A  could  do     —     of  the  work,  and 

B,    —  :     hence,  by  the  conditions  of  the  question 
lo 


KT  13       ' 
which  gives  1  3x  +  1  Ox  =  1  30  : 

hence,  23x=  130,     x=—  —  =  5£|     days. 

«o 

12.  A  fox,  pursued  by  a  greyhound,  has  a  start  of  60 
leaps.  He  makes  9  leaps  while  the  greyhound  makes  but 
6  ;  but,  three  leaps  of  the  greyhound  are  equivalent  to  7 
of  the  fox.  How  many  leaps  must  the  greyhound  make  to 
overtake  the  fox  ? 

From  the  enunciation,  it  is  evident  that  the  distance  to 
be  passed  over  by  the  greyhound  is  composed  of  the  60 
leaps  which  the  fox  is  in  advance,  plus  the  distance  that  the 
fox  passes  over  from  the  moment  when  the  greyhound  starts 
in  pursuit  of  him.  Hence,  if  we  can  find  the  expression  for 
these  two  distances,  it  will  be  easy  to  form  the  equation  of 
the  problem. 

Let  x=  the  -number  of  leaps  made  by  the  greyhound 
before  he  overtakes  the  fox. 

Now,  since  the  fox  makes  9  leaps  while  the  greyhound 

9  3 

makes  but  6,"  the  fox  will  make     —     or     —     leaps  while 

6  2 


86  ELEMENTARY  ALGEBRA. 

the  greyhound  makes  1  ;  and,  therefore,  while  the  greyhound 

3 
makes  x  leaps,  the  fox  will  make     —  #     leaps. 

2 

Hence,  the  distance  which  the  greyhound  must  pass  over 

3 

will  be  expressed  by     60  -\  --  x     leaps  of  the  fox. 

2 

It  might  be  supposed,  that  in  order  to  obtain  the  equation, 

3 
it  would  be  sufficient  to  place  x  equal  to     60+—  z;     but 

in  doing  so,  a  manifest  error  would  be  committed  ;  for  the 
leaps  of  the  greyhound  are  greater  than  those  of  the  fox,'and 
we  should  then  equate  numbers  of  different  denominations  ; 
that  is,  numbers  referring  to  different  units.  Hence  it  is 
necessary  to  express  the  leaps  of  the  fox  by  means  of  those 
of  the  greyhound,  or  reciprocally.  Now,  according  to  the 
enunciation,  3  leaps  of  the  greyhound  are  equivalent  to  7 
leaps  of  the  fox,  then  1  leap  of  the  greyhound  is  equiva- 

lent to     —     leaps  of  the  fox,  and  consequently  x  leaps  of 
o 

7x 

the  greyhound  are  equivalent  to     —     of  the  fox. 

o 

Hence,  we  have  the  equation 


making  the  denominators  disappear 


whence  .  5#—  360     and     x=72. 

Therefore  the  greyhound  will  make   72  leaps  to  overtake 
the  fox,  and  during  this  time  the  fox  will  make 

72  X          or     108 


EQUATIONS    OF    THE    FIRST    DEGREE.  87 

Verification. 
The  72  leaps  of  the  greyhound  are  equivalent  to 


leaps  of  the  fox.     And 

60+108  =  168, 

the  leaps  which  the  fox  made  from  the  beginning. 

13.  A  father  leaves  his  property,  amounting  to  $2520,  to 
four  sons,  A,  B,  C,  and  D.     C  is  to  have  $360,  B  as  much 
as  C  and  D  together,  and  A  twice  as  much  as   B   less 
$1000  :  how  much  does  A,  B,  and  D  receive  ? 

Ans.  A  $760,  B  $880,  D  $520. 

14.  An  estate  of  $7500  is  to  be  divided  between  a  widow, 
two  sons,  and  three  daughters,  so  that  each  son  shall  receive 
twice  as  much  as   each  daughter,  and  the  widow  herself 
$500  more  than  all  the  children  :  what  was  her  share,  and 
what  the  share  of  each  child  ? 

/  Widow's  share  $4000. 
Ans.  J  Each  son's  $1000. 

t  Each  daughter's  $500. 

15.  A  company  of  180  persons  consists  of  men,  women, 
and  children.     The  men  are  8  more  in  number  than  the 
women,  and  the  children  20  more  than  the  men  and  women 
together  :  how  many  of  each  sort  in  the  company  ? 

Ans.  44  men,  36  women,  100  children. 

16.  A  father  divides  $2000  among  five  sons,  so  that  each 
elder  should  receive   $40  more  than  his  next  younger  bro- 
ther: what  is  the  share  of  the  youngest?  Ans.  $320. 

17.  A  purse  of  $2850  is  to  be  divided  among  three  per- 
sons, A,  B,  and  C.     A's  share  is  to  be  to  B's  as  6  to  1  1. 


88  ELEMENTARY  ALGEBRA. 

and  C  is  to  have  $300  more  than  A  and  B  together :  what 
is  each  one's  share  ? 

Ans.  A's  $450,  B's  $825,  C's  $1575 

18.  Two  pedestrians  start  from  the  same  point;  the  first 
steps  twice  as  far  as  the  second,  but  the  second  makes  5 
steps  while  the  first  makes  but  one.     At  the  end  of  a  certain 
time  they  are  300  feet  apart.     Now,  allowing  each  of  the 
longer  paces  to  be  3  feet,  how  far  will  each  have  travelled? 

Ans.   1st,  200 /e^;  2nd,  500. 

19.  Two  carpenters,  24  journeymen,  and  8  apprentices, 
received  at  the  end  of  a  certain  time  $144.     The  carpen- 
ters received   $1   per  day,  each  journeyman  half  a  dollar, 
and  each  apprentice  25  cents :  how  many  days  were  they 
employed  ?  A  ns.  9  a1  ays. 

20.  A  capitalist  receives  a  yearly  income  of  $2940  :  four- 
fifths  of  his  money  bears  an  interest  of  4  per  cent,  and  the 
remainder  at  5  per  cent :  how  much  has  he  at  interest  ? 

Ans.  70000. 

21.  A  cistern  containing  60  gallons  of  water  has  three 
unequal  cocks  for  discharging  it ;  the  largest  will  empty  it 
in  one  hour,  the  second  in  two  hours,  and  the  third  in  three  : 
in  what  time  will  the  cistern  be   emptied  if  they  all  run 
together  ?  AJIS.  32^min. 

22.  In  a  certain  orchard  ±  are  apple  trees,  J  peach  trees, 
J  plum  trees,  120  cherry  trees,  and   80  pear  trees  :  how 
many  trees  in  the  orchard  ?  Ans.  2400. 

23.  A  farmer   being    asked    how    many  sheep  he  had, 
answered  that  he  had  them  in  five  fields  ;  in  the  1  st   he 
had  i,  in  the  2nd  J,  in  the  3rd  J,  and  in  the  4th  j^,  and  in 
the  5th  450  :  how  many  had  he  ?  Ans.   1200. 

24.  My  horse  and  saddle  together  are  worth  $132,  and 
the  horse  is  worth  ten  times  as  much  as  the  saddle  :  what 
is  the  value  of  the  horse  ?  Ans.   120. 


EQUATIONS    OF    THE    FIRST    DEGREE.  8<) 

25.  The  rent  of  'm  estate  is  this  year  8  per  cent  greater 
than  it  was  last.     This  year  it  is  $1890  :  what  was  it  last 
year?  Ans.   $1750. 

26.  What  number  is  that  from  which,  if  5  be  subtracted, 
jj  of  the  remainder  will  be  40  ?  Ans.  65. 

27.  A  post  is   1  in  the  mud,  J  in  the  water,  and  10  feet 
above  the  water :  what  is  the  whole  length  of  the  post  ? 

Ans.  24  feet. 

28.  After  paying  \  and  \  of  my  money,  I  had  66  guineas 
left  in  my  purse  :  how  many  guineas  were  in  it  at  first  ? 

Ans.  120. 

29.  A  person  was  desirous  of  giving  3  pence  apiece  to 
some  beggars,  but  found  he  had  not  money  enough  in  his 
pocket  by  8  pence :  he  therefore  gave  them  each  2  pence 
and  had  3  pence  remaining  :  required  the  number  of  beggars. 

Ans.   11. 

30.  A  person  in  play  lost  J  of  his  money,  and  then  won 
3   shillings ;  after  which  he  lost  J  of  what  he   then  had ; 
and  this  done,  found  that  he  had  but  12  shillings  remaining  : 
what  had  he  at  first  ?  Ans.  20.?. 

31.  Two  persons,  A  and  B,  lay  out  equal  sums  of  money 
in  trade;  A  gains  $126,  and  B  loses  $87,  and  A's  money 
is  now  double  of  B's  :  what  did  each  lay  out  ? 

Ans.  $300. 

32.  A  person  goes  to  a  tavern  with  a  certain  sum  of  mo- 
ney in  his  pocket,  where  he  spends  2  shillings  ;  he  then 
borrows  as  much  money  as  he  had  left,  and  going  to  another 
tavern,  he  there  spends  2  shillings  also ;  then  borrowing 
again  as  much  money  as  was  left,  he  went  to  a  third  tavern, 
where  likewise  he  spent  two  shillings   and  borrowed  as 
mirch  as  he  had  left ;  and  again  spending  2  shillings  at  a 
fourth  tavern,  he  then  had  nothing  remaining.     What  had 
he  ai  first?  Ans.   3*.  9rf. 

9 


90  ELEMENTARY  ALGEBRA. 


Of  Equations  of  the  First  Degree  involving  two  or 
more  unknown  quantities. 

72.  Although  several  of  the  questions  hitherto  resolved 
contained  in  their  enunciation  more  than  one  unknown  quan- 
tity, we  have  resolved  them  all  by  employing  but  one  sym- 
bol. The  reason  of  this  is,  that  we  have  been  able,  from 
the  conditions  of  the  enunciation,  to  express  easily  the  other 
unknown  quantities  by  means  of  this  symbol ;  but  we  are 
unable  to  do  this  in  all  problems  containing  more  than  one 
unknown  quantity. 

To  ascertain  how  problems  of  this  kind  are  resolved,  let 
us  take  some  of  those  which  have  been  resolved  by  means 
of  one  unknown  quantity. 

1 .  Given  the  sum  of  two  numbers  equal  to  36  and  their 
difference  equal  to  12,  to  find  the  numbers. 

Let  x=  the  greater,  and  y=  the  less  number. 

Then,  by  the  1st  condition x+y— 36, 

and  by  the  2nd, x— y  =  12. 

By  adding  (Art.  65,  Ax.  1),       ....  2#=48. 

By  subtracting  (Art.  65,  Ax.  2),     ...  2y=24. 

Each  of  these  equations  contains  but  one  unknown  quantity 

46 

From  the  first  we  obtain #=—=24. 

24 
And  from  the  second y=——I2 

Verification. 

ar+y=36     gives     24+12  =  36 
x— V=12        „         24  —  12  =  12 


2 
a-b 


EQUATIONS    OF    THE    FIRST    DEGREE.  91 

General  Solution. 
Let  x=.  the  greater,  and  y  the  less  number. 

Then  by  the  conditions x-{-y=.a, 

and         x—y=b. 

By  adding,  (Art.  65,  Ax.  1),     .     .     .  2x=a+b. 

By  subtracting,  (Art.  65,  Ax.  2),    .     .  2y=a—b. 

Each  of  these  equations  contains  but  one  unknown  quantity 
From  the  first  we  obtain x= 

And  from  the  second y= 

'* 

Verification. 

a+b  ,  a  —  b     2a  a+b      a—b     2b 

— +-5- =T=«;     and    —    — =y=*. 

For  a  second  example,  let  us  also  take  a  problem  that  has 
oeen  already  solved. 

2.  A  person  engaged  a  workman  for  48  days.  For  each 
day  that  he  labored  he  was  to  receive  24  cents,  and  for  each 
day  that  he  was  idle  he  was  to  pay  12  cents  for  his  board. 
At  the  end  of  the  48  days  the  account  was  settled,  when 
the  laborer  received  504  cents.  Required  the  number  of 
working  days,  and  the  number  of  days  he  was  idle. 

Let  x=.         the  number  of  working  days, 

y=         the  number  of  idle  days. 
Then,         24a?=r         what  he  earned, 
and  12y=         what  he  paid  for  his  board. 

Then,  by  the  conditions  of  the  question,  we  have 

x+y      =48, 
and  24#—  12y  =  504. 

This  is  the  statement  of  the  question. 


92  ELEMENTARY  ALGEBRA. 

It  has  already  been  shown  (Art.  65,  Ax.  3),  that  the  two 
members  of  an  equation  can  be  multiplied  by  the  same  num- 
ber, without  destroying  the  equality.  Let,  then,  the  first 
equation  be  multiplied  by  24,  the  coefficient  of  x  in  the 
second  :  we  shall  then  have 


24a—  12y—   504. 
And  by  subtracting,  36y=   648, 

648 
and  y=~W=18' 

Substituting  this  value  of  y  in  the  equation 

24a>—  12y  =  504,     we  have     24#—  216  =  504, 
which  gives 

24*=504-f-216:=720,     and     x=^-  =  30. 

Verification. 

x+     y=  48     gives  30  +  18=   48, 

24*—  12y=504     gives     24  x  30-12  x  18  =  504. 

Elimination. 

7  3.  The  method  which  has  just  been  explained  of  com- 
bining two  equations,  involving  two  unknown  quantities,  and 
deducing  therefrom  a  single  equation  involving  but  one,  is 
called  elimination. 


QUEST. — 73.  What  is  elimination  1  How  many  methods  of  ehm;na- 
tion  are  there  1  Give  the  rule  for  elimination  by  addition  and  subtrac- 
tion t  What  is  the  first  step]  What  the  second]  What  the  third' 


EQUATIONS    OF    THE    FIRST    DEGREE.  93 

There  are  three  principal  methods  of  elimination  : 

1st.  By  addition  and  subtraction 

2d.    By  substitution. 

3d.    By  comparison. 
We  will  consider  these  methods  separately. 

Elimination  by  Addition  and  Subtraction. 

1.  Take  the  two  equations 

3x-2y=7 


If  we  add  these  two  equations,  member  to  member,  we 
obtain 


which  gives,  by  dividing  by  11 


and  substituting  this  value  in  either  of  the  given  equations, 
we  find 


2.  Again,  take  the  equations 

8a:-h2y-48 
3x+2y=23. 

If  we  subtract  the  2nd  equation  from  the  first,  we  obtain 

5x=25, 
which  gives,  by  dividing  by  5 

x=5: 
and  by  substituting  this  value,  we  find 


9* 


94  ELEMENTARY  ALGEBRA. 

3    Take  the  two  equations 


If,  in  these  equations,  one  of  the  unknown  quantities  was 
affected  with  the  same  coefficient,  we  might,  by  a  simple 
subtraction,  form  a  new  equation  which^would  contain  but 
one  unknown  quantity. 

Now,  if  both  members  of  the  first  equation  be  multiplied 
by  9,  the  coefficient  of  y  in  the  second,  and  the  two  mem- 
bers of  the  second  by  7,  the  coefficient  of  y  in  the  first,  we 
will  obtain 

45z+63y=387, 
77o?+63y=483. 

Subtracting,  then,  the  first  of  these  equations  from  the 
second,  there  results 

32#=96,     whence     x=3. 

Again,  if  we  multiply  both  membeis  of  the  first  equation 
by  1  1  ,  the  coefficient  of  x  in  the  second,  and  both  members 
of  the  second  by  5,  the  coefficient  of  x  in  the  first,  we  will 
form  the  two  equations 

55;r+77y  =  473, 


Subtracting,  then,  the  second  of  these  two  equations  from 
the  first,  there  results 

32y=128,     whence     y=4. 
Therefore  x  =3  and  y—  4,  are  the  values  of  x  and  y. 

Verification* 

5;r+7y=43     gives       5  x  3  +  7x4  =  15+28=43  ; 
ll*+95=69  11x3  +  9x4= 


EQUATIONS    OF    THE    FIRST    DEGREE.  95 

The  method  of  elimination  just  explained,  is  called  the 
method  by  addition  and  subtraction.- 

To  eliminate  by  this  method  we  have  the  following 

RULE. 

.  I.  See  which  of  the  unknown  quantities  you  will  eliminate. 

II.  Make  the  coefficient  of  this  unknown  quantity  the  same 
in  both  equations,  either  by  multiplication  or  division. 

III.  If  the  signs  of  the  like  terms  are  the  same  in  both 
equations,  subtract  one  equation  from   the  other  ;  but  if  the 
signs  are  unlike,  add  them. 

EXAMPLES. 

4.  Find  the  values  of  x  and  y  in  the  equations 

3*—  y=3, 
y+2x=7. 

Ans.  x=2,  y=3. 

5.  Find  the  values  of  x  and  y  in  the  equations 

4x—  7y=  —22, 
5a?  |-2y=37. 

Ans.  x=5t  y=6 

6.  Find  the  values  of  x  and  y  in  the  equations 


8x—6y=  3. 

Ans.  x=4%,  y=5|. 
7.  Find  the  values  of  a?  and  y  in  the  equations 

8x—9y=l. 
6x—3y  =  4x. 

Ans.  x=,  y~. 


96  ELEMENTARY  ALGEBRA. 

8.  Find  the  values  of  .r  and  y  in  the  equations 

14*  —  15y  —  12, 
7x+   8y  =  37. 

Ans.  x  =3,  y=%> 

9.  Find  the  values  of  x  and  y  in  the  equations 


10.  Find  the  values  of  x  and  y  in  the  equations 

1«+1,=4, 

a—  y—  —  2. 

Ans.  x=14,  y-16. 

11.  Says  A  to  B,  you  give  me  $40  of  your  money,  and 
I  shall  then  have  5  times  as  much   as  you  will  have  left. 
Now  they  both  had  $120  :  how  much  had  each  ? 

Ans.  Each  had  $60. 

12.  A   Father  says  to  his  son,  "twenty  years  ago,  my 
age  was  four  times  yours  ;  now,  it  is  just  double  ;"  what  wer 
their  ages  ?  4          5  Father's  60  years 

\  Son's       30  years. 

13.  A  Father  divides  his  property  between  his  two  sons. 
At  the  end  of  the  first  year  the  elder  had  spent  one  quarter 
of  his,  and  the  younger  had  made  $1000,  and  their  property 
was  then  equal.     After  this  the  elder  spends  $500  and  the 
younger  makes   $2000,  when  it  appears  the  younger  has 
just  double  the  elder  :  what  had  each  from  the  father  ? 

.          (  Elder        $4000 
t  Younger  $2000 


EQUATIONS    OF    THE    FIRST    DEGREE.  97 

14.  If  John  give  Charles  15  apples,  they  will  have  the 
same  number;  but  if  Charles  give  15  to  John,  John  will 
have   15  times  as  many  wanting  10  as  Charles  will  have 
left.     How  many  had  each  ?  .         (  John       50. 

(Charles  20. 

15.  Two  clerks,  A  and  B,  have  salaries  which  are  to 
gether  equal  to  $900.     A  spends  ^  per  year  of  what  he 
receives,  and  B  adds  as  much  to  his  as  A  spends.     At  the 
end  of  the  year  they  have  equal  sums  :  what  was  the  salary 
of  each? 


'Elimination  by  Substitution. 
7  4.  Let  us  again  take  the  equations 

5a?+7y=43, 
Ila:-h9y=69. 

Find  the  value  of  x  in  the  first  equation,  which  gives 


Substitute  this  value  of  x  in  the  second  equation,  and  we 
have 

43— 7y 
11  X r-^-+9y=69, 

D 

or, 
or, 
Hence, 

and, 


98  ELEMENTARY  ALGEBRA. 

This  method  is  called  the  method  by  substitution :  we 
have  for  it  the  following 

RULE. 

Find  the  value  of  one  of  the  unknown  quantities  in  either 
of  the  equations,  and  substitute  this  value  for  the  same  unknown 
quantity  in  the  other  equation :  there  will  thus  arise  a  new 
equation  with  but  one  unknown  quantity. 

REMARK. — This  method  of  elimination  is  used  to  great 
advantage  when  the  coefficient  of  either  of  the  unknown 
quantities  is  unity. 

•it    ' 

EXAMPLES. 
/ 

1 .  Find,  by  the  last  method,  the  values  of  x  and  y  in  the 
equations 

3a?— y=l     and     3y— 2a?=4 

Ans.  «=1,  y=2. 

2.  Find  the  values  of  x  and  y  in  the  equations 

5y— 4a?=  —22     and     3y-f4a?=38. 

Ans.  x—8,  y=2. 

3.  Find  the  values  of  x  and  y  in  the  equations 

a?+8y=18     and     y— 3x=—  29. 

Ans.  x=lQ,  y=l. 

4.  Find  the  values  of  x  and  y  in  the  equations 

5*-y=13   and   8x+^-y=29. 

y 
Ans.  #— 3J,  y=4j. 

QUEST. — 74.  Give  the  rule  for  elimination  by  substitution'  When  is 
it  desirable  to  use  this  method11 


EQUATIONS    OF    THE    FIRST    DEGREE.  99 

6.  Find  the  values  of  x   and   y   from  the  equations 

IQx—  ^-=69     and     lOy—  ^-=49. 
o  7 

Ans.  #=7,  y=5. 
6.  Find  the  values  of  x   and  y   from  the  equations 

-=I°  and         - 


Ans.  a?=8,  y=10. 
7.  Find  the  values  of  x  and   y   in  the  equations 


8    Find  the  values  of  x  and   y   in  the  equations 
i+     +.=          ana    ±-±. 


9.  Find  the  values  of  x  and   y   from  the  equations 


10.  Find  the  values  of  a   and   y   from  the  equations 

--i—'  and  5^-=29- 


?=6,  y=7. 

1  1  .  Two  misers  A  and  B  sit  down  to  count  over  their 
money.  They  both  have  $20000,  and  B  has  three  times  as 
much  as  A  :  how  much  has  each  ? 

A  .  .      $5000. 


B  .       $15000 


100  ELEMENTAKV  ALGEBRA. 

12.  A  person  has  two  purses.     If  he  puts   $7  into  the 
first  purse,  it  is  worth  three  times  as  much  as  the  second  : 
but  if  he  puts  $7  into  the  second  it  becomes  worth  five 
times  as  much  as  the  first  :  what  is  the  value  of  each  purse  ? 

Ans.  1st,  $2  :  2nd,  $3. 

13.  Two  numbers  have  the  following  properties  :    if  the 
first  be   multiplied  by  6  the  product  will  be  equal  to  the 
second  multiplied  by  5  ;  and  one  subtracted  from  the  first 
leaves  the  same  remainder  as  2  subtracted  from  the  second  : 
what  are  the  numbers  ?  Ans.  5  and  6. 

14.  Find  two  numbers  with  the    following   properties  : 
the   first  increased  by  2  to  be  3£  times  greater  than  the 
second  :  and  the  second  increased  by  4  to  be  half  the  first  : 
what  are  the  numbers  ?  Ans.  24  and  8. 

15.  A  father  says  to  his  son,  "  twelve  years  ago  I  was 
twice  as  old  as  you  are  now  :  four  times  your  age,  at  that 
time,  plus  twelve  years,  will  express  my  age  twelve  years 
hence  :"  what  were  their  ages  ?        ^       (  Father  72  yeyrs. 

nS'  \ 


Son       30 


Elimination  by  Comparison. 
75.  Take  the  same  equations 


llx+9y=69. 
Finding  the  value  of  x  in  the  first  equation,  we  have 

_43-7y  . 
~5~ 

and  finding  the  value  of  x   in  the  second,  we  obtain 

x=     TT~' 


EQUATIONS    OF    THE    FIRST    DEGREE.  101 

Let  these  two  values  of  x  be  placed  equal  to  each  other, 
and  we  have 

43—  7y  _  69   -9y 
5  ~~~ll      ' 


Or, 

473  —  77y  =  34£ 

)—  45y; 

Or, 

—  32y=  —  128. 

Hence. 

y=4. 

And, 

69  —  36 

=  3. 

This  method   of  elimination   is    called  the    method  by 
comparison,  for  which  we  have  the  following 


RULE. 

I.  Find  the  value  of  the  same  unknown  quantity  in  each 

equation. 

II.  Place  these  values  equal   to  each  other;  and   a    new 
equation  will  arise  with  but  one  unknown  quantity. 

EXAMPLES. 

1  .  Find,  by  the  last  rule,  the  values  of  x   and   y   in  the 
equations 

JL_|_6=42     and     y—^-=U\. 
o  22 

Ans.  x=ll,  y=15 


QUEST. — 75.  Give  the  rule  for  elimination  by  comparison  ?     What  is 
th<  first  stop 7     What  the  second  1 

10 


102  ELEMENTARY  ALGEBRA. 

2.  Find  the  values  of  x   and   y  in  the  equations 

l_f+5=6     and     f  +4=^+6. 

Ans.  a:—  28,  y=20. 

3.  Find  the  values  of  x  and  y  in  the  equations 

v       x      22 

-+=1    and    3-*=6' 


4.  Find  the  values  of  x   and   y   in  the  equations 

y-3  =  l*+5     and     fltlUy-S}. 

Ans.  #=2,  y=9. 

5.  Find  the  values  of  x   and   y   in  the  equations 


.  ar=16,  y  =  7. 
6.  Find  the  values  of  a;   and   y   from  the  equations 

y  +  x     y—x  2y 

^—  f~  -=x—+,     and     a:+y=16. 


.  a;=10,  y=6 
7.  Find  the  values  of  x  and  y  in  the  equations 


8.  Find  the  values  of  x  and  y   in  the  equations 

2y+3*=y+43,    y-^^-y- 
Ans.  x=W, 


EQUATIONS    OF    THE    FIRST    DEGREE.  103 

9.  Find  the  values  of  x  and  y  in  the  equations 
4y—x^y=x-^l8l   and   27—  y=ff-fy+4. 

Ans.  a?=9,     y=7 
1.0.  Find  the  vralues  of  x  and  y  in  the  equations 


)  . 

Ans.  ff=10,     y=20. 

76.  Having  explained  the  principal  methods  of  elimina- 
tion, we  shall  add  a  few  examples  which  may  be  solved  by 
either  ;  and  often  indeed,  it  may  be  advantageous  to  use 
them  all  even  in  the  same  question. 

GENERAL    EXAMPLES. 

1.  Given     2o?+3y=16,    and    3x—  2y=ll    to  find  the 
values  of  x  and  y.  Ans.  x=5,   y=2. 

2x  ,  3y       9          ,    3x     2y       61 

2.  Given    -+JL  =-   and  -^+i=l^  to  find  the 

values  of  x  and  y.  Ans.  x=  —  ,   y=  —  . 

3.  Given     ~-{-7y=99,    and   -|--f  7a?=51,    to  find  the 
values  of  x  and  y.  Ans.  x  =7,   y=14. 

4.  Given 


-12=+8,   and 


to  find  the  values  of  x  and  y.  Ans,  x  =  6Q,   y  —  40. 


104  ELEMENTARY  ALGEBRA. 


QUESTIONS. 

1.  What  fraction  is  that,  to  the  numerator  of  which,  if  ] 
be  added,  its  value  will  be     — ,     but  if  one  be  added  to  its 

o 

denominator,  its  value  will  be     —  ? 

4 

Let  the  fraction  be  represented  by     — . 


Then,  by  the  question 


a+1       I  .         x          I 

— • — = —     and      = — . 

y         3  y+1       4 

Whence  3x+3  =  y     and     4a?=y+l. 

Therefore,  by  subtracting, 

x—3  =  l     or     oc=4. 
Hence,  12  +  3=y; 

therefore,  y  — 15. 

2.  A  market-woman  bought  a  certain  number  of  eggs  ai 
2  for  a  penny,  and  as  many  others,  at  3  for  a  penny ;  and 
having  sold  them  again  altogether,  at  the  rate  of  5  for  2d, 
found  that  she  had  lost  4d :  how  many  eggs  had  she  ? 

Let  2o?=     the  whole  number  of  eggs. 

Then  x=     the  number  of  eggs  of  each  sort. 

Then  will        "9-*=     the  cost  °f  tne  ^rst  sort> 

and  — x=     the  cost  of  the  second  sort. 

o 

But     5  :  2x  :  :    2    :  —     the  amount  for  which  the   eggs 
o 

wore  sold. 


EQUATIONS    OF    THE    FIRST    DEGREE.  105 

Hence,  by  the  question 

11         4x 
_*+_*__  =  4. 

Therefore,  15x+Wx—24x=120 

or  a::=120 ; 

the  number  of  eggs  of  each  sort. 

3.  A  person  possessed  a  capital  of  30,000  dollars,  foi 
which  he  drew  a  certain  interest ;  but  he  owed  the  sum  of 
20,000  dollars,  for  which  he  paid  a  certain  interest.  The 
interest  that  he  received  exceeded  that  which  he  paid  by 
800  dollars.  Another  person  possessed  35,000  dollars,  for 
which  he  received  interest  at  the  second  of  the  above  rates  ; 
but  he  owed  24,000  dollars,  for  which  he  paid  interest  at 
the  first  of  the  above  rates.  The  interest  that  he  received 
exceeded  that  which  he  paid  by  310  dollars.  Required  the 
two  rates  of  interest. 

Let  oc  and  y  denote  the  two  rates  of  interest ;  that  is,  the 
interest  of  $100  for  the  giv#n  time. 

To  obtain  the  interest  of  $30,000  at  the  first  rate,  denoted 
by  a?,  we  form  the  proportion 

100  :  x  :  :  30,000  :  :     3°f°°X      or     300*. 
100 

And  for  the  interest  $20,000,  the  rate  being  y, 

100  :  y  :  :  20,000  :  :     ™g™S.     or     200y. 

But  from  the  enunciation,  the  difference  between  these 
i  wo  interests  is  equal  to  800  dollars. 

We  have,  then,  for  the  first  equation  of  the  problem 

300*— 200y— 800 
10* 


106  ELEMENTARY  ALGEBRA. 

By  writing  algebraically  the  second  condition  of  the  pro- 
blem, we  obtain  the  other  equation, 


Both  members  of  the  first  equation  being  divisible  by  1  00, 
and  those  of  the  second  by  10,  we  may  put  the  following, 
in  place  of  them  : 

3x—  2   =  8          35 


To  eliminate  x,  multiply  the  first  equation  by  8,  and  then 
add  it  to  the  second  ;  there  results 

19y=95,     whence     y=5. 

Substituting  for  y,  in  the  first  equation,  its  value,  this 
equation  becomes 

3x—  10  —  8,     whence     x=6. 
Therefore,  the  first  rate  is  6  per  cent,  and  the  second  5. 

Verification. 

$30,000,     placed  at  6  per  cent,  gives     300x6  —  $1800. 
$20,000,          „  5       „  „         200x5  =  $1000. 

And  we  have  1800  —  1  000  —  800. 

The  second  condition  can  be  verified  in  the  same  manner. 

4.  What  two  numbers  are  those,  whose  difference  is  7, 
and  sum  33  1  Ans.   13  and  20. 

5.  To  divide  the  number  75  into  two  such  parts,  that 
three  times  the  greater  may  exceed  seven  times  the  less 
by  15.  Ans.  54  and  21. 

6.  In  a  mixture  of  wine  and  cider,  J  of  the  whole  plus 
25  gallons  was  wine,  and  J  part  minus  5  gallons  was  cider  : 
how  many  gallons  were  there  of  each  ? 

Ans.  85  of  wine,  and  35  of  cider 


EQUATIONS    OF    THE    FIRST    DEGREE.  107 

7.  A  bill  of  £120  waa.paid  in  guineas  and  moidores,  and 
the  number  of  pieces  of  both  sorts  that  were  used  was  just 
100.     If  the  guinea  be  estimated  at  21*,  and  the  moidore 
at  27s,  how  many  were  there  of  each  ?      Ans.  50  of  each. 

8.  Two  travellers  set  out  at  the  same  time  from  London 
and  York,  whose  distance  apart  is  150  miles.     One  of  them 
goes  8  miles  a  day,  and  the  other  7  :  in  what  time  will  they 
meet?  Ans.  In  10  days. 

9.  At  a  certain  election,  375  persons  voted  for  two  can- 
didates, and  the  candidate  chosen  had  a  majority  of  91  : 
how  many  voted  for  each  ? 

Ans.  233  for  one,  and  142  for  the  other. 

10.  A  person  has  two  horses,  and  a  saddle  worth  £50. 
Now,  if  the  saddle  be  put  on  the  back  of  the  first  horse,  it 
will  make  his  value  double  that  of  the  second ;  but  if  it  be 
put  on  the  back  of  the  second,  it  will  make  his  value  triple 
that  of  the  first.     What  is  the  value  of  each  horse  ?     . 

Ans.  One  .£30,  and  the  other  £40. 

1 1 .  The  hour  and  minute  hands  of  a  clock  are  exactly 
together  at  12  o'clock  :  when  are  they  next  together  ? 

Ans.   Ihr.  5T5Tmm. 

12.  A  man  and  his  wife  usually  drank  out  a  cask  of  beer 
in  12  days  ;  but  when  the  man  was  from  home,  it  lasted 
the  woman  30  days :  how  many  days  would  the  man  alone 
be  in  "drinking  it?  Ans.  20  days. 

13.  If  32  pounds  of  sea-water  contain  1  pound  of  salt, 
how  much  fresh  water  must  be  added  to  these  32  pounds, 
in  order  that  the  quantity  of  salt  contained  in  32  pounds  of 
the  new  mixture  shall  be  reduced  to  2  ounces,  or  l  of  a 
pound?  Ans.  224lb. 

14.  A  person  who  possessed  100,000  dollars,  placed  the 
greater  part  of  it  out  at  5  per  cent  interest,  and  the  other 


1  08  ELEMENTARY  ALGEBRA. 

at  4  per  cent.     The  interest  which  he  received  for  tr    whole 
amounted  to  4640  dollars.     Required  the  two  parts. 

Ans.  64,000  arid  36,000. 

15.  At  the  close  of  an  election,  the  successful  candidate 
had  a  majority  of  1500  votes.     Had  a  fourth  of  the  votes 
of  the  unsuccessful  candidate  been  also  given  to  him,  he 
would  have  received  three  times  as  many  as  his  competitor 
wanting  three  thousand  five  hundred  :  how  many  votes  did 
each  receive?  (  1st,  6500 

<  2d,    5000. 

16.  A  gentlemen  bought  a  gold  and  a  silver  watch,  and 
a  chain  worth  $25.     When  he  put  the  chain  on  the  gold 
watch,  it  was  worth  three  and  a  half  times  more  than  the 
silver  watch  ;  but  when  he  put  the  chain  on  the  silver  watch, 
it  was  worth  one  half  the  gold  watch  and  15  dollars  over: 
what  was  the  value  of  each  watch  ? 

Gold  watch  $80. 


Ans. 

Silver     „      $30. 

17.  There  is  a  certain  number  expressed  by  two  figures, 
which  figures  are  called  digits.     The  sum  of  the  digits  is 
11,  and  if  13  be  added  to  the  first  digit  the  sum  will  be  three 
times  the  second  :  what  is  the  number  ?  Ans.  56. 

18.  From  a  company  of  ladies  and  gentlemen  15  ladies 
retire ;    there  are  then  left  two  gentlemen   to   each   lady. 
After  which,  45  gentlemen  depart,  when  there  are  left  5 
ladies  to  each  gentleman  :  how  many  were  there  of  each  at 
first?  A          (50  gentlemen. 

(  40  ladies. 

19.  A  person  wishes  to  dispose  of  his  horse  by  lottery 
If  he  sells  the  tickets  at  $2  each,  he  will  lose  $30  on  his 
horse  ;  but  if  he  sells  them  at  $3  each,  he  will  receive  $30 
more  than  his  horse  cost  him.     What  is  the  value  of  the 
horse  and  number  of  tickets?       A__     $  Horse    .  .  .  $150. 

its     60 


(  Horse    .  .  . 
Ans.    < 

(No.  of  ticket 


EQUATIONS    OF    THE    FIRST    DEGREE.  109 

20.  A  person  purchases  a  lot  of  wheat  at  $1 ,  and  a  lot  of 
rye  at  75  cents  per  bushel,  the  whole  costing  him  $117,50. 
He  then  sells  £  of  his  wheat  and  £  of  his  rye  at  the  same 
rate,  and  realizes  $27,50.  How  much  did  he  buy  of  each  ? 

(  80bu.  of  wheat 
Ans.    < 

50bu.  of  rye. 


Equations  involving  three  or  more  unknown  quantities. 

77.  Let  us  now  consider  the  case  of  three  equations 
involving  three  unknown  quantities. 
Take  the  equations 


—  3z=  19, 


To  eliminate  z  by  means  of  the  first  two  equations,  mul- 
tiply the  first  by  3  and  the  second  by  4  ;  then,  since  the 
coefficients  of  z  have  contrary  signs,  add  the  two  results 
>,ogether.  This  gives  a  new  equation  : 

43ar—  2y=121. 

Multiplying  the  second  equation  by  2,  a  factor  of  the  co- 
efficient of  z  in  the  third  equation,  and  adding  them  together, 
we  have 

16a;-f9y  =  84. 

The  question  is  then  reduced  to  finding  the  values  of  x 
and  y,  which  will  satisfy  these  new  equations. 

Now,  if  the  first  be  multiplied  by  9,  the  second  by  2,  and 
the  results  be  added  together,  we  find 

whence     x=3 


110  ELEMENTARY  ALGEBRA. 

We  might,  by  means  of  the  two  equations  involving  a 
and  y,  determine  y  in  the  same  way  we  have  determined  x  ; 
but  the  value  of  y  may  be  determined  more  simply,  by  ob- 
serving that  the  last  of  these  two  equations  becomes,  by 
substituting  for  x  its  value  found  above, 

48-f-9y=84,     whence    y= — - — =4. 

y 

In  the  same  manner  the  first  of  the  three  proposed  equa- 
tions becomes,  by  substituting  the  values  of  x  and  y, 

24 
15 — 24-{-4z  =  l5,     whence     z=—  =  6. 

4 

Hence,  to  solve  equations  containing  three  or  more  un- 
known quantities,  we  have  the  following 

RULE. 

I.  To  eliminate  one  of  the  unknown  quantities,  combine  any 
one  of  the  equations  with  each  of  the  others ;  there  will  thus  be 
obtained  a  series  of  new  equations  containing  one  less  unknown 
quantity. 

II.  Eliminate  another  unknown  quantity  by  combining  one 
of  these  new  equations  with  the  others. 

III.  Continue  this  series  of  operations  until  a  single  equa- 
tion containing  but  one  unknown  quantity  is  obtained,  from 
which  the  value  of  this  unknown   quantity  is  easily  found. 
Then,  by  going  back  through  the  series  of  equations  which  have 
been  obtained,  the  vdtues  of  the  other  unknown  quantities  may 
be  successively  determined. 

QUEST. — 77.  Give  the  general  rule  for  solving  equations  involving 
three  or  more  unknown  quantities'!  What  is  the  first  step]  What  the 
second  1  What  the  third  ? 


EQUAT1OJNS    OF    THE    FIRST    DEGREE.  Ill 

78.  REMARK.  —  It  often  happens  that  each  of  the  pro- 
posed equations  does  not  contain  all  the  unknown  quantities. 
In  this  case,  with  a  little  address,  the  elimination  is  very 
quickly  performed. 

Take  the  four  equations  involving  four  unknown  quantities: 


(1.)     2x—  3y+2z=  13.  (3.) 

(2.)  4u—  2x=  30.  (4.)      5y+3«=32. 

By  inspecting  these  equations,  we  see  that  the  elimina- 
tion of  z  in  the  two  equations,  (1)  and  (3),  will  give  an 
equation  involving  x  and  y  ;  and  if  we  eliminate  u  in 
the  equations  (2)  and  (4),  we  shall  obtain  a  second  equation, 
involving  x  and  y.  These  two  last  unknown  quantities 
may  therefore  be  easily  determined.  In  the  first  place,  the 
elimination  of  z  in  (1)  and  (3)  gives 

7y—2x=l  ; 
That  of  u   in  (2)  and  (4),  gives 


Multiplying  the  first  of  these  equations  by  3,  and  adding, 

41y=41  ; 

Whence  y=    I. 

Substituting  this  value  in    7y—  2x=l,  we  find 

x=3. 
Substituting  for  x  its  value  in  equation  (2),  it  becomes 

4w  —  6  —  30  : 

Whence  u  =  9. 

And  substituting  for    y   its  value  in  equation  (3),  there 
results  z  =5. 


1.  Given 


3.   Given 


ELEMENTARY  ALGEBRA. 


EXAMPLES. 


—x-\ yH z  =  \5  }>  to  find  a?,  y  and  z. 

3          45 


Ans.     a?=12,  y=r20,  ^  = 


4.  Divide  the  number  90  into  four  such  parts  that  tht 
first  increased  by  2,  the  second  diminished  by  2,  the  third 
multiplied  by  2,  and  the  fourth  divided  by  2,  shall  be  equa\ 
to  each  other. 

This  question  may  be  easily  solved  by  introducing  a  new 
unknown  quantity. 

Let  a?,  y,  z,  and  u,  be  the  required  parts,  and  desig 
nate  by  m  the  several  equal  quantities  which  arise  from 
the  conditions.  We  shall  then  have 


=m. 


EQUATIONS    OF    THE    FINEST    DEGREE.  113 

Krom  which  we  find 

x=.m — 2,    y=rm-f-2,    z——  ,   «ac2m. 
And  by  adding  the  equations, 

x  +  y+z  +  u— .m-\-m-\ — —  4-2m=4jm. 

And  since,  by  the  conditions  of  the  question,  the  first 
member  is  equal  to  90,  we  have 

41/72  =  90,     or     fm  =  90; 
hence  m=20. 

Having  the  value  of  m,  we  easily  find  the  other  values  : 
viz. 

*  =18,   y=22,    *=10,    «  =  40. 

5.  There  are  three  ingots  composed  of  different  metals 
mixed  together.  A  pound  of  the  first  contains  7  ounces  of 
silver,  3  ounces  of  copper,  and  6  of  pewter.  A  pound  of 
the  second  contains  12  ounces  of  silver,  3  ounces  of  cop- 
per, and  1  of  pewter,  A  pound  of  the  third  contains  4 
ovnces  of  silver,  7  ounces  of  copper,  and  5  of  pewter.  It 
is  required  to  find  how  much  it  will  take  of  each  of  the 
three  ingots  to  form  a  fourth,  which  shall  contain  in  a 
pound,  8  ounces  of  silver,  3^  of  copper,  and  4^-  of  pewter. 

Let  ar,  y,  and  z  represent  the  number  of  ounces  which  it 
is  necessary  to  take  from  the  three  ingots  respectively,  in 
order  to  form  a  pound  of  the  required  ingot.  Since  there 
are  7  ounces  of  silver  in  a  pound,  or  16  ounces,  of  the 
first  ingot,  it  follows  that  one  ounce  of  it  contains  r7^  of  an 
ounce  of  silver,  and  consequently  in  a  number  of  ounce* 

denoted  by  a?,  there  is    — -   ounces  of  silver.      In  the  sam« 
lo 

11 


I  14  ELEMENTARY  ALGEBRA. 


manner  we  would  find  that   —  2-  and  —  ,   express  the  num 

16  16 

her  of  ounces  of  silver  taken  from  the  second  and  third,  to 
form  the  fourth  ;  but  from  the  enunciation,  one  pound  of  this 
fourth  ingot  contains  8  ounces  of  silver.  We  have,  then, 
for  the  first  equation, 

7x      I2y      4* 
16+    16   +  16- 

or,  making  the  denominators  disappear, 


As  respects  the  copper,  we  should  find 
3tf  +  3y  +7*  =  60, 
and  with  reference  to  the  pewter 


As  the  coefficients  of  y  in  these  three  equations,  are 
the  most  simple,  it  is  most  convenient  to  eliminate  this  un- 
known quantity  first. 

Multiplying  the  second  equation  by  4,  and  subtracting 
the  first,  we  have 

5x  +  24z=:ll2. 

Multiplying  the  third  equation  by  3,  and  subtracting  the 
second  from  the  product, 


Multiplying  this  last  equation  by  3,  and  subtracting  the 
preceding  one  from  the  product,  we  obtain 

40*:=  320, 
whence  x=S. 


EQUATIONS    OF    THE    FIRST    DEGREE. 

Substitute  this  value  for   x   in  the  equation 


it  becomes  120  +  8*  =  144, 

whence  z=3. 

Lastly,  the  two  values    a;  =8,  z=3,   being  substituted  in 
the  equation 


give  48-hy-fl5=68, 

whence  y=5- 

Therefore,  in  order  to  form  a  pound  of  the  fourth  ingot, 
we  must  take  8  ounces  of  the  first,  5  ounces  of  the  second, 
and  3  of  the  third. 

Verification. 

If  there  be  7  ounces  of  silver  in  1  6  ounces  of  the  first 
ingot,  in  8  ounces  of  it,  there  should  be  a  number  of  ounces 
of  silver  expressed  by 

7x8 
16  ' 
In  like  manner, 

12x5  4x3 

and 


16  16 

will  express  the  quantity  of  silver  contained  in  5  ounces  of 
the  second  ingot,  and  3  ounces  of  the  third. 
Now,  we  have 

7X8      12x5      4x3_  128  _ 
16        ~~I6~~        16    "=    16    ~     ' 

therefore,  a  pound  of  the  fourth  ingot  contains  8  ounces  of 
silver,  as  required  by  the  enunciation.  The  same  condi- 
tions may  be  verified  relative  to  the  copper  and  pewter 


1  1  6  ELEMENTARY  ALGEBRA. 

6    A's  age  is  double  B's,  and  B's  is  triple  of  C's,  and  the 
sum  of  all  their  ages  is  140.     What  is  the  age  of  each  ? 

Ans.  A's~84,  B's  =  42,  and  C's=14. 

7.  A  person  bought  a  chaise,  horse,  and  harness,  for 
£60  ;  the  horse  came  to  twice  the  price  of  the  harness, 
and  the  chaise  to  twice  the  price  of  the  horse  and  harness 
What  did  he  give  for  each  ? 

£.£13     6s.  8d.     for  the  horse. 
Ans.    <  .£  6  13s.  4d.     for  the  harness 
'  £40  for  the  chaise. 

8.  To  divide  the  number  36  into  three  such  parts  that 
£  of  the  first,  i  of  the  second,  and  J  of  the  third,  may  be 
all  equal  to  each  other.  Ans.  8,  12,  and  16. 

9.  If  A  and  B  together  can  do  a  piece  of  work  in  8  days, 
A  and  C  together  in  9  days,  and  B  and  C  in  ten  days  ;  how 
many  days  would  it  take  each  to  perform  the  same  work 
alone?    "  Ans.  A  14f|,  B  17|f,  C  23^T. 

10.  Three  persons,  A,  B,  and  C,  begin  to  play  together, 
having  among  them  all  $600.     At  the  end  of  the  first  game 
A  has  won  one-half  of  B's  money,  which,  added  to  his  own, 
makes  double  the  amount  B  had  at  first.     In  the   second 
game,  A  loses  and  B  wins  just  as  much  as  0  had  at  the 
beginning,  when  A  leaves  off  with  exactly  what  he  had  at 
first.     How  much  had  each  at  the  beginning  ? 

Ans.  A  $300,  B  $200,  C  $100. 

11.  Three  persons,  A,  B,  and  C,  together  possess  $3640. 
If  B  gives  A  $400  of  his  money,  then  A  will  have  $320 
more  than  B  ;  but  if  B  takes   $140  of  C's  money,  then  B 
and  C  will  have  equal  sums.     How  much  has  each  ? 

Ans.  A  $800,  B  $1280,  C  $1560. 

12.  Three  persons  have   a  bill  to  pay,  which  neither 
alone  is  able  to  discharge.     A  says  to  B,  "  Give  me  the 
4th  of  your  money,  and  then  I  can  pay  the   bill."     B  says 
to  C,  "  Give  me  the  8th  of  yours,  and  I  can  pay  it.     Bui 


EQUATIONS    OF    THE    FIRST    DEGREE.  1  1 7 

C  says  to  A,  "  You  must  give  me  the  half  of  yours  before 
I  can  pay  it,  as  I  have  but  $8."  What  was  the  amount  of 
their  bill,  and  how  much  money  had  A  and  B  ? 

.         1  Amount  of  the  bill,  $13. 
'   \  A  had  $10,  and  B  $12. 

13.  A  person  possessed  a  certain  capital,  which  he  placed 
out  at  a  certain  interest.     Another  person,  who  possessed 
10000  dollars  more*  than  the  first,  and  who  put  out  his  capi- 
tal 1  per  cent,  more  advantageously,  had  an  income  greater 
by  800  dollars.     A  third  person,  who  possessed  15000  dol- 
lars more  than  the  first,  putting  out  his  capital  2  per  cent, 
more  advantageously,  had  an  income  greater  by  1 500  dollars. 
Required  the  capitals  of  the  three  persons,  and  the  rates  of 
interest. 

,        <  Sums  at  interest,     $30000,     40000,     4500U. 
(  Rates  of  interest,  4  5  6  pr.  ct. 

14.  A  widow  receives  an  estate  of  $15000  from  her  de- 
ceased husband,  with  directions  to  divide  it  among  two  sons 
and  three  daughters,  so  that  each  son  may  receive  twice  as 
much  as  each  daughter,  and  she  herself  to  receive  $1000 
more  than  all  the  children  together.     What  was  her  share, 
and  what  the  share  of  each  child  ? 

^  The  widow's  share,  $8000. 

Ans.  ?  Each  son's,  2000. 

'  Each  daughter's,          1000. 

15.  A  certain  sum  of  money  is  to  be  divided  between 
three  persons,  A,  B,  and  C.     A  is  to  receive  $3000  less 
than  half  of  it,  B  $1000  less  than  one  third  part,  and  C  to 
receive  $800  more  than  the  fourth  part  of  the  whole.     What 
is  the  sum  to  be  divided,  and  what  does  each  receive  ? 

Sum,  $38400. 

A  receives  16200. 
B  „  11800. 
C  10400. 


11  fc  ELEMENTARY  ALGEHRA. 


CHAPTER    IV. 

Of  Powers. 

79.  If  a  quantity  be  multiplied  several  times  by  itself 
the  product  is  called  the  power  of  the  quantity.  Thus, 

a  =  a  is  the  root,  or  first  power  of  a. 

axa  =  a2  is  the  square,  or  second  power  of  a. 

aXaXa=a3  is  the  cube,  or  third  power  of  a. 

aXaXaXa=a*  is  the  fourth  power  of  a. 

aXaXaXaX a=a5  is  the  fifth  power  of  a. 

In  every  power  there,  are  three  things  to  be  considered  : 

1st.  The  quantity  which  is  multiplied  by  itself,  and  which 
is  called  the  root  or  the  first  power. 

2nd.  The  small  figure  which  is  placed  at  the  right,  and 
a  little  above  the  letter.  This  figure  is  called  the  exponent 
of  the  power,  and  shows  how  many  times  the  letter  enters 
as  a  factor. 

3rd.  The  power  itself,  which  is  the  final  product,  or 
result  of  the  multiplications. 


QUEST. — 79.  If  a  quantity  be  continually  multiplied  by  itself,  what  is 
the  product  called  1  How  many  things  are  to  be  considered  in  every 
power  1  What  are  they  1 


OF    POWERS.  119 

For  example,  if  we  suppose  <z  =  3,  we  h?ve 

o—     3     the  root,  or  1st  power  of  3. 


uz—32=3x3=     9  the  second  power  of  3. 

a*  =  33  =  3  x3x3=  27  the  third  power  of  3. 

a*  =  3*  —  3  x3  x3x3=:   81  the  fourth  power  of  3. 

=  35=:3x  3x3x3x3=243  the  fifth  power  of  3. 


In  these  expressions,  3  is  the  root,  1,  2,  3,  4  and  5  are 
the  exponents,  and  3,  9,  27,  81  and  243  are  the  powers. 

To  raise  monomials  to  any  power. 

8O.  Let  it  be  required  to  raise  the  monomial  2a?b2  to 
the  fourth  power.     We  have 


which  merely  expresses  that  the  fourth  power  is  equal  to 
the  product  which  arises   from  writing  the  quantity  four 
times  as  a  factor.     By  the  rules  for  multiplication,  this  pro 
duct  becomes 

=  24a3  +  3  +  3  +  362  +  2  +  2  +  2  =  24a1268  ; 

from  which  we  see, 

1st.  That  the  coefficient  2  must  be  raised  to  the  4th 
power  ;  and, 

2nd.  That  the  exponent  of  each  letter  must  be  multiplied 
by  4,  the  exponent  of  the  power. 

As  the  same  reasoning  would  apply  to  every  example, 
we  have,  for  the  raising  of  monomials  to  any  power,  the 
following 


120  ELEMENTARY  ALGEBRA. 

RULE 

I.  Raise  the  coefficient  to  the  required  power. 

II.  Multiply  the  exponent  of  each  letter  by  the  exponent  oj 
the  power. 

EXAMPLES. 

1.  What  is  the  square  of   3a2y3  ?  Ans.  9a4/ 

2.  What  is  the  cube  of   6a5y2a?  ?  Ans.  216a15y«a:3 

3.  What  is  the  fourth  power  of  2a3y3i5  ? 


4.  What  is  the  square  of  «2A5y3?  4ns.  </4i10y6. 

5.  What  is  the  seventh  power  of  a2bcd?  ? 

4ns.  a1*  We'd?1. 

6.  What  is  the  sixth  power  of  aWczdl     Ans.  a12£18c12</6. 

7.  What  is  the  square  and  cube  of    —2a2b2  1 

Square.  Cube. 


By  observing  the  way  in  which  the  powers  are  formed, 
we  may  conclude, 

1st.    When  the  root  is  positive,  all  the  powers  will  be  positive. 

2nd.  When  the  root  is  negative,  all  the  even  powers  will  be 
positive  and  all  the  odd  powers  negative. 

QUEST.  —  8O.  What  is  a  monomial  1  Give  the  rule  for  raising  a 
monomial  to  any  power.  When  the  root  is  positive,  how  will  the  powers 
be  1  When  the  root  is  negative,  how  will  the  powers  he  ' 


OF    POWERS.  121 

8.  What  is  the  square  of   —  2<z455  ?  Ans.  4a*bw. 

9.  What  is  the  cube  of  —5a5y2c  ?        Ans.   —  125al*yec3. 

10.  What  is  the  eighth  power  of    —a3xy2  1 

Ans.  +a24a?8y16 

11.  What  is  the  seventh  power  of    —azyx2  ? 

Ans.   —  aWyV*: 

12.  What  is  the  sixth  power  of  2ab*y5  ? 

Ans.  64a*b36y30. 

13.  What  is  the  ninth  power  of   —  cdx2y3  1 

Ans.   — 

14.  What  is  the  sixth  power  of   —  3ab2d  ? 

Ans. 

15.  What  is  the  square  of   —WaWc3  ?      Ans.  100a464c6. 

16.  What  is  the  cube  of    —  9a<W3/2  ? 

Ans.   — 

17.  What  is  the  fourth  pow.er  of   —  4a5b3c*d5  ? 


18.  What  is  the  cube  of    —4a2b2c3d  ? 

Ans.   - 

19.  What  is  the  fifth  power  of  2a3b*xy  ? 

Ans. 

20.  What  is  the  square  of  20a;4y4c5  ?        Ans.  400x8y8c}0. 

21.  What  is  the  fourth  power  of  3azb2c3  ? 

Ans. 

22.  What  is  the  fifth  power  of  —c^x2^  ? 

Ans.   — 

23.  What  is  the  sixth  power  of  —ac2df1 

Ans.  « 

24.  What  is  the  fourth  power  of  —  2a2c2d3  ? 

Ans. 


1  22 


ELEMENTARY  ALGEBRA. 


To  raise  Polynomials  to  any  power. 

8  1  .  The  power  of  a  polynomial,  like  that  of  a  mono- 
mial, is  obtained  by  multiplying  the  quantity  continually  by 
itself.  Thus,  to  find  the  fifth  power  of  the  binomial  a-\  b, 
we  have 


a  4-  b 

a  +  b 

a2+  ab 

4- 


az+2ab 
a  4-   b 


1st  power. 


2nd  power. 


ab* 


4- 


4-   b3 


a34-3a2&4- 
a  +  b 


-f 


.     3rd  power. 


-f   &4     4th  powey 


a  + 


6a362-j- 


4- 


a362  +  1  Oa<263  +  5a 


Ans. 


REMARK.  —  82.  It  will  be  observed  that  the  number  of 
multiplications  is  always  1  less  than  the  units  in  the  expo- 


QUEST.  —  81.  How  is  the  power  of  a  polynomial  obtained  ? 


OF    POWERS. 


123 


nent  of  the  power.  Thus,  if  the  exponent  is  1,  no  multipli- 
cation is  necessary.  If  it  is  2,  we  multiply  once  ;  if  it  is 
3,  twice  ;  if  4,  three  times,  &c.  The  powers  of  polyno- 
mials may  be  expressed  by  means  of  the  exponent.  Thus, 
lo  express  that  a+b  is  to  ba  raised  to  the  5th  power,  we  write 


which  expresses  the  fifth  power  of  a-\-b. 
2.  Find  the  5th  power  of  the  binomial  a  —  b. 

a  —  b     ........     1st  power. 

a  -  b 
a2  —  ab 

-   ab  +6* 
a?—2ab  -{-b2     .          ....     2nd  power. 

a  -     b 


a3—2azb  +      abz 


a3—  3a2b+   3ab2  —b3      ...     3rd  power. 

a  -  b 


—  ab3 

—  3ab3 


6a2&2—  4ab3  +   b*    4th  power. 


a  -   b 


—  4a2b3+   ab* 


a5— 


—  b5     Ans. 


QUEST.  —  82.  How  does  the  number  of-  multiplications  compare  with 
the  exponent  of  the  power1?  If  the  exponent  is  4,  how  many  multipli- 
cations 1 


124 


ELEMENTARY  ALGEBRA. 


3.  What  is  the  square  of    5a— 

5a  —  2c  +     d 

5a  —  2c  +     d 

25a2—  10ac+   5ad 

—  10<zc+   4c2—  2cd 


25«2— 


—4cd+cP     Ans. 


4.  Find  the  4th  power  of  the  binomial     3a  —  2b. 

3a  —     25  .     .     .     .     .     .     .     .     1st  power. 


9a2—     6ab 


9a2— 

3a  —     2b 


2nd  power. 


27  a3—   36a*b+12ab* 


27a3— 
3a  —     2b 


.     .     3rd  power. 


Sla4— 


An*. 


5.  What  is  the  square  of  the  binomial  a-\-  1  ? 

Ans.  a 

6.  What  is  the  square  of  the  binomial  a  —  1  ? 

Ans.  a*—2a+l. 

7.  What  is  the  cube  of  9a—  36? 

Ans.  729a3—  729a25+243«62  —  2753 

8.  What  is  the  third  power  of  a—  11 

Ans.  a3  —  3a2-f3a—  1. 


OF    POWERS.  125 


9    What  is  the  4th  power  of  x—yl 
Ans.  x* 


10.  What  is  the  cube  of  the  trinomial  x-\-y-\-z1 

Ans.  x3  +  3o%  +  3x2z  -f  3xf  +  3xz2  +  3y*z  -f  3y*2  +  Gays 


11.  What  is  the  cube  of  the  trinomial  2a2—  4aZ>-f  3b2  ? 
.  8a6  — 


To  raz^e  a  Fraction  to  any  Power. 

83.  The  power  of  a  fraction  is  obtained  by  multiplying 
the  fraction  by  itself  ;  that  is,  by  multiplying  the  numerator 
by  the  numerator,  and  the  denominator  by  the  denominator 

Thus,  the  cube  of  -=-,    which  is  written 
o 


a  a       a       a 


is  found  by  cubing  the  numerator  and  denominator  sepa- 
rately. 

2.  What  is  the  square  of  the  fraction     -r  -  ? 

b-\-c 

We  have 

/a—  c\2_  (a—cf  _  at—Zac+c2 
~  -~ 


3.  What  is  the  cube  of    --  ?  Ans. 


.  •- 

3  be  27b3c3 


QUEST. — 83     How  do  you  find  the  power  of  a  fraction 1 
12 


ELEMENTARY  ALGEBRA. 


4.  What  is  the  fourth  power  of     —  — 

*<"t> 


5.  What  is  the  cube  of 


^' 

2  ax 


»>    What  is  the  fourth  power  of 

7.  \Vhat  is  the  fifth  power  of 

8.  What  is  the  square  of 


9.  What  is  the  cube  of  2 


Binomial  Theorem. 


16aV 
x— v  „ 


84.  The  method  which  has  been  explained  of  raising  a 
polynomial  to  any  power,  is  somewhat  tedious,  and  hence 
other  methods,  less  difficult,  have  been  anxiously  sought 
for.  The  most  simple  which  has  yet  been  discovered,  is 
the  one  invented  by  Sir  Isaac  Newton,  called  the  Binomial 
Theorem. 


QUEST._84.    What  is  the  object  of  the  Binomial  Theorem  1     Who 
discovered  this  theorem  1 


BINOMIAL    THEOREM.  127 

85.  In  raising  a  quantity  to  any  power,  it  is  plain  that 
there  are  four  things  to  be  considered  :  — 

1st.  The  number  of  terms  of  the  power. 
2nd.  The  signs  of  the  terms. 
3rd.  The  exponents  of  the  letters. 
4th.  The  coefficients  of  the  terms. 

Of  the  Terms. 

86.  If  we  take  the  two  examples  of  Article  81,  which 
we  there  wrought  out  in  full  ;  we  have 


By  examining  the  several  multiplications,  in  Art.  8  1  ,  we 
shall  observe  that  the  second  power  of  a  binomial  contains 
three  terms,  the  third  power  four,  the  fourth  power  five,  the 
fifth  power  six,  &c  ;  and  hence  we  may  conclude  —  That 
the  number  of  terms  in  any  power  of  a  binomial,  is  one  greater 
than  the  exponent  $f  the  power. 

Of  the  Signs  of  the  Terms. 

87.  It  is  evident  that  when  both  terms  of  the  given  bi- 
nomial are  plus,  all  the  terms  of  the  power  will  be  plus. 

2nd.  If  the  second  term  of  the  binomial  is  negative,  then 
all  the  odd  terms,  counted  from  the  left,  will  be  positive,  and 
all  the  even  terms  negative. 


QUEST. — 85.  In  raising  a  quantity  to  any  power,  how  many  things 
are  to  be  considered  ?  What  are  they  1 — 86.  How  many  terms  are 
there  in  any  power  of  a  binomial  ?  If  the  exponent  is  3,  how  many 
terms'!  If  it  is  4,  how  many  terms  I  If  5]  &c. — 87.  If  both  terms 
of  the  binomial  are  positive,  how  are  the  terms  of  the  power?  If  the 
second  term  is  negative,  how  are  the  signs  of  the  terms  1 


128  ELEMENTARY  ALGEBRA. 

Of  the  Exponents. 

88.  The  letter  which  occupies  the  first  place  in  a  bino- 
mial, is  called  the  leading  letter.  Thus,  a  is  the  leading 
letter  in  the  binomials  a-\-b,  a—b. 

1st.  It  is  evident  that  the  exponent  of  the  leading  lettei 
in  the  first  term  will  be  the  same  as  the  exponent  of  the 
power  ;  and  that  this  exponent  will  diminish  by  unity  in 
each  term  to  the  right,  until  we  reach  the  last  term,  which 
does  not  contain  the  leading  letter. 

2nd.  The  exponent  of  the  second  letter  is  1  in  the  second 
term,  and  increases  by  unity  in  each  term  to  the  right 
until  we  reach  the  last  term,  in  which  the  exponent  is  the 
same  as  that  of  the  given  power. 

3rd.  The  sum  of  the  exponents  of  the  two  letters,  in  any 
term,  is  equal  to  the  exponent  of  the  given  power.  This 
last  remark  will  enable  us  to  verify  any  result  obtained  by 
the  binomial  theorem. 

Let  us  now  apply  these  principles  in  the  two  following 
examples,  in  which  the  coefficients  are  omitted  :  — 


(a  —  b)6  .  .  .  a6—  «5&-f  a4//—  a?b*+a2b4—  a 

As  the  pupil  should  be  practised  in  writing  the  terms, 
with  their  proper  signs,  without  the  coefficients,  we  will  add 
a  few  more  examples. 


QUEST. — 88.  Which  is  the  leading  letter  of  the  binomial  1  What  is 
the  exponent  of  this  letter  in  the  first  term  1  How  does  it  change  in  the 
terms  towards  the  right  1  What  is  the  exponent  of  the  second  letter  in 
the  second  term?  How  does  it  change  in  the  terms  towards  the  right  1 
What  is  it  in  the  last  term7  What  is  the  sum  of  the  exponents  in  any 
term  equal  to7 


BINOMIAL     THEOREM. 


2.   (a—*)4  .   .  «4-a 

3. 

4.  (a  — 


O/"  £/*e  Coefficients. 

89.  The  coefficient  of  the  first  term  is  unity.  The  co- 
efficient of  the  second  term  is  the  same  as  the  exponent  of 
the  given  power.  The  coefficient  of  the  third  term  is  found 
by  multiplying  the  coefficient  of  the  second  term  by  the 
exponent  of  the  leading  letter,  and  dividing  the  product  by 
2.  And  finally  —  If  the  coefficient  of  any  term^be  multiplied 
by  the  exponent  of  the  leading  letter,  and  the  product  divided 
by  the  number  which  marks  the  place  of  that  term  from  the 
left,  the  quotient  will  be  the  coefficient  of  the  next  term. 

Thus,  to  find  the  coefficients  in  the  example 

(a  —  I)1   .  .  .  a'  —  a*b  +  a5b*  —  a*y>  \  a3b*-a?b5-\-al6-  V 

we  first  place  the  exponent  7  as  a  coefficient  of  the  second 
term.  Then,  to  find  the  coefficient  of  the  third  term,  we 
multiply  7  by  6,  the  exponent  of  a,  and  divide  by  2.  The 
quotient  21  is  the  coefficient  of  the  third  term.  To  find  the 
coefficient  of  the  fourth,  we  multiply  21  by  5,  and  divide 
the  product  by  3  :  this  gives  35.  To  find  the  coefficient  of 
the  fifth  term,  we  multiply  35  by  4,  and  divide  the  product 
by  4  :  this  gives  35.  The  coefficient  of  the  sixth  term, 
found  in  the  same  way,  is  21  ;  that  of  the  seventh,  7  ;  and 
that  of  the  eighth,  1.  Collecting  these  coefficients,  we 
have 


-  7aGb  +  2  1  a562  - 

12* 


I  30  ELEMENTARY  ALGEBRA. 

REMARK. — We  see,  in  examining  this  last  result,  that  the 
coefficients  of  the  extreme  terms  are  each  unity,  and  that 
the  coefficients  of  terms  equally  distant  from  the  extreme 
terms  are  equal.  It  will,  therefore,  be  sufficient  to  find  the 
coefficients  of  the  first  half  of  the  terms,  from  which  the 
others  may  be  immediately  written. 

j  EXAMPLES. 

1.  Find  the  fourth  power  of  a+b. 

Ans. 

2.  Find  the  fourth  power  of  a— b. 

Ans.   a4— 

3.  Find  the  fifth  power  of  a+b. 

Ans.  «5  +  5a4&+1 

4.  Find  the  fifth  power  of  a  —  b. 

Ans.   a5  —  5a4&+10a3#>-10a2&3+5flM— b5. 

5.  Find  the  sixth  power  of  a-\-b. 

Ans.  «6+6a55+15a4i2+20a3^3+15a2^+6a55-f-^. 

6.  Find  the  sixth  power  of  a  —  b. 

Ans.  a6  —  6a56+15a462— 20«3634-15a2£4— 6ab5+b* 

7.  Let  it  be  required  to  raise  the  binomial    3a2c—2bd  to 
the  fourth  power. 

It  frequently  occurs  that  the  terms  of  the  binomial  are 
affected  with  coefficients  and  exponents,  as  in  the  above 


QUEST — 89.  What  is  the  coefficient  of  the  first  term?  What  is  the 
coefficient  of  the  second  1  How  do  you  find  the  coefficient  of  the  third 
term  1  How  do  you  find  the  coefficient  of  any  term  1  What  are  the 
coefficients  of  the  first  and  last  terms  1  How  are  the  coefficients  o/ 
terms  equally  distant  from  the  extremes  1 


BINOMIAL    THEOREM.  131 

example.     In  the  first  place,  we  represent  each  term  of  the 
binomial  by  a  single  letter.     Thus,  we  place 

3a2c=x,     and     —2bd=i/, 
we  then  have 


But,  x2  =  9a*c2,     x3=27a6c3,     aj4  = 

and  y2=4bW,y3=  —  8b3d?,     y* 

Substituting  for    x    and    y    their  values,  we  have 

(3a3c—  2W)*  =  (3a2c)*+4(3a2c)3(—  2bd)  +  6  (3a2c)2  (—  2bd)z 
+  4(3a2c)  (-2bd)3  +  (-2bd)\ 

and  by  performing  the  operations  indicated, 

—  2l6a6c3bd  +  2l6a*cWd?— 


8.  What  is  the  square  of   3a  —  6b  ? 

Ans.  9«2  — 

9.  What  is  the  cube  of     3x—  6y  T 

Ans.     27x3  —  l62x2y+324xy* 

10.  What  is  the  square  of    x—y  ? 

Ans.  x2— 

11.  What  is  the  eighth  power  of  m-\-n  ? 
Ans. 


12.  What  is  the  fourth  power  of     a—  3b  ? 

Ans.  a*  —  l2a3b+54azb2— 

13.  What  is  the  fifth  power  of     c—  2d  1 

Ans.  c5  —  1  Oc4d  +  40c3d2  — 

14.  What  is  the  cube  of    5a  —  3d  ? 

Ans.  I25a3—  225a?d+l35ad2—  27d3 


1  32  ELEMENTARY  ALGEBRA. 

REMARK.    The  powers  of  any  polynomial  may  easily  he 
found  by  the  Binomial  Theorem. 

15.  For  example,  raise    a  -\-b-\-c    to  the  third  power 

First,  put    ....    b-\-c—  d 
Then,     (a+£  +  e)3  =  (a  +  </)3  = 
Or.  by  substituting  for  the  value  of   d, 


This  expression  is  composed  of  the  cubes  of  the  three 
terms,  plus  three  times  the  square  of  each  term  by  the  first 
powers  of  the  two  others,  plus  six  times  the  product  of  alt 
three  terms.  It  is  easily  proved  that  this  law  is  true  for  any 
polynomial. 

To  apply  the  preceding  formula  to  the  development  of 
the  cube  of  a  trinomial,  in  which  the  terms  are  affected 
with  coefficients  and  exponents,  designate  each  term  by  a 
single  letter,  then  replace  the  letters  introduced,  by  their  values, 
and  perform  the  operations  indicated. 

From  this  rule,  we  find  that 


(2a2  —  4ab  +  3b*)3  =  8a6-48a5b+132aW—  208a3b3 
+  1  98a2M  -  1  QSab*  +  27  b6. 

The  fourth,  fifth,  &c,  powers  of  any  polynomial  can  be 
found  in  a  similar  manner. 

16.  What  is  the  cube  of     a—  2b+c  ? 
Ans.  «3~8Z/3  +  c3 
—  6bc2—  I2abc. 


EXTRACTION   OP  THE  SQUARE  ROOT. 


CHAPTER   V. 

Extraction  of  the  Square  Roof  of  Numbers.  Forma" 
tion  of  the  Square  and  Extraction  of  the  Square 
Root  of  Algebraic  Quantities.  Calculus  of  Radicals 
of  the  Second  Degree. 

90.  The  square  or  second  power  of  a  number,  is  the 
product  which  arises  from  multiplying  that  number  by  itself 
once  :   for  example,  49  is  the  square  of  7,  and   144  is  the 
square  of  12. 

9 1 .  The  square  root  of  a  number  is  that  number  which, 
being  multiplied  by  itself  once,  will  produce  the  given  num- 
ber.    Tnus,  7  is  the  square  root  of  49,  and   12  the  square 
root  of  144:    for,  7x7  =  49,  and   12x12  =  144. 

92.  The  square  of  a  number,  either  entire  or  fractional, 
is  easily  found,  being  always  obtained  by  multiplying  this 
number  by  itself  once.     The  extraction  of  the  square  root 
of  a  number  is,  however,  attended  vyith  some  difficulty,  and 
requires  particular  explanation. 


QUEST. — 90.    What  is  the  square,  or  second  power  of  a  number? — 
01 .    What  is  the  square  root  of  a  number  ^ 


134  .        ELEMENTARY  ALGEBRA. 

The  first  ten  numbers  are. 

1,     2,     3,       4,       5,       6,       7,       8,       9,       10; 
and  their  squares, 

1,     4,     9,     16.     25,     36,     49,     64,     81,     100; 

and  reciprocally,  the  numbers  of  the  first  line  are  the  square 
roots  of  the  corresponding  numbers  of  the  second.  We 
may  also  remark  that,  the  square  of  a  number  expressed  by  a 
single  figure,  will  contain  no  figure  of  a  higher  denomination 
than  tens. 

The  numbers  of  the  last  line,  1,  4,  9,  16,  &c,  and  all 
other  numbers  which  can  be  produced  by  the  multiplication 
of  a  number  by  itself,  are  called  perfect  squares. 

It  is  obvious  that  there  are  but  nine  perfect  squares  among 
all  the  numbers  which  can  be  expressed  by  one  or  two 
figures  :  the  square  roots  of  all  other  numbers  expressed 
by  one  or  two  figures,  will  be  found  between  two  whole 
numbers  differing  from  each  other  by  unity.  Thus  55, 
which  is  comprised  between  49  and  64,  has  for  its  square 
root  a  number  between  7  and  8.  Also  91,  which  is  com- 
prised between  81  and  100,  has  for  its  square  root  a  number 
between  9  and  10. 

93.  Every  number  may  be  regarded  as  made  up  of  a 
certain  number  of  tens  and  a  certain  number  of  units. 
Thus  64  is  made  up  of  6  tens  and  4  units,  arid  may  be  ex- 
pressed under  the  form  60  +  4. 


QUEST. — 92.  What  will  be  the  highest  denomination  of  the  square 
of  a  number  expressed  by  a  single  figure  1  What  are  perfect  squares  1 
How  many  are  there  between  1  and  100  1  What  are  they  ? 


EXTRACTION    OF    THE    SQUARE    ROOT.  135 

Now,  if  we  represent  the  tens  by  a  and  the  units  by  b, 
we  shall  have 

a+b    =  64, 
and 


or 

Which  proves  that  the  square  of  a  number  composed  of 
tens  and  units,  contains  the  square  of  the  tens  plus  twice  the 
product  of  the  tens  by  the  units,  plus  the  square  of  the  units. 

94.  If,  now,  we  make  the  units  1,2,  3,  4,  &c,  tens,  or 
units  of  the  second  order,  by  annexing  to  each  figure  a  ci- 
pher, we  shall  have 

10,    20,    30,      40,      50,      60,      70,      80,       90,       100, 
and  for  their  squares, 
100,  400,  900,  1600,  2500,  3600,  4900,  6400,  8100,  10000. 

From  which  we  see  that  the  square  of  one  ten  is  100,  the 
square  of  two  tens  400  ;  and  generally,  that  the  square  of 
tens  will  contain  no  figures  of  a  less  denomination  than  hun- 
dreds, nor  of  a  higher  name  than  thousands. 

Ex.  1.  —  To  extract  the  square  root  of  6084. 

Since  this  number  is  composed  of  more  than 
two  places  of  figures,  its  root     will   contain  60  84 

more  than  one.     But  since  it  is  less  than  1  0000, 
which  is  the  square  of  100,  the  root  will  contain  but  two 
figures  :  that  is,  units  and  tens. 

Now,  the  square  of  the  tens  must  be  found  in  the  two 


QUEST. — 93.  How  may  every  number  be  regarded  as  made  up  f  What 
is  the  square  of  a  number  composed  of  tens  and  units  equal  to1!— 
94.  What  is  the  square  of  one  ten  equal  to  ?  Of  2  tens  1  Of  3 
tens '  &c. 


136 


ELEMENTARY  ALGEBRA. 


left-hand  figures,  which  we  will  separate  from  the  other  two 
by  putting  a  point  over  the  place  of  units,  and  a  second  over 
the  place  of  hundreds.  These  parts,  of  two  figures  each,  are 
called  periods.  The  part  60  is  comprised  between  the  two 
squaies  49  and  64,  of  which  the  roots  are  7  and  8  :  hence, 
7  is  the  figure  of  the  tens  sought ;  and  the  required  root  is 
composed  of  7  tens  and  a  certain  number  of  units. 

The  figure  7  being  found,  we 
write  it  on  the  right  of  the  given 
number,  from  which  we  separate 
it  by  a  vertical  line  :  then  we 
subtract  its  square,  49,  from  60, 
which  leaves  a  remainder  of  1 1 , 
to  which  we  bring  down  the  two 

next  figures  84.  The  result  of  this  operation,  1184,  con- 
tains twice  the  product  of  the  tens  by  the  units,  plus  the  square 
of  the  units. 

But  since  tens  multiplied  by  units  cannot  give  a  product  of 
a  less  name  than  tens,  it  follows  that  the  last  figure,  4,  can 
form  no  part  of  the  double  product  of  the  tens  by  the  units : 
this  double  product  is  therefore  found  in  the  part  118,  which 
we  separate  from  the  units'  place,  4. 

Now  if  we  double  the  tens,  which  gives  14,  and  then  di- 
vide 118  by  14,  the  quotient  8  is  the  figure  of  the  units,  or 
a  figure  greater  than  the  units.  This  quotient  figure  can 
never  be  too  small,  since  the  part  118  will  be  at  least  equal 
to  twice  the  product  of  the  tens  by  the  units  :  but  it  may  be 
too  large ;  for  the  118,  besides  the  double  product  of  the 
tens  by  the  units,  may  likewise  contain  tens  arising  from 
the  square  of  the  units.  To  ascertain  if  the  quotient  8  ex- 
presses the  units,  we  write  the  8  on  the  right  of  the  14, 
which  gives  148,  and  then  we  multiply  148  by  8.  Thus, 
we  evidently  form,  1st,  the  square  of  the  units;  and, 
2nd,  the  double  product  of  the  tens  by  the  units.  Thi* 


EXTRACTION  OF  THE  SQUARE  ROC  f.      137 

multiplication  being  effected,  gives  for  a  product  1184,  a 
number  equal  to  the  result  of  the  first  operation.  Having 
subtracted  the  product,  we  find  the  remainder  equal  to  0  : 
hence  78  is  the  root  required. 

Indeed,  in  the  operations,  we  have  merely  subtracted 
from  the  given  number  6084,  1st,  the  square  of  7  tens,  or 
70  ;  2nd,  twice  the  product  of  70  by  8  ;  and,  3d,  the  square 
of  8  :  that  is,  the  three  parts  which  enter  into  the  composi- 
tion of  the  square  70 -(-8,  or  78  ;  and  since  the  result  of 
the  subtraction  is  "0,  it  follows  that  78  is  the  square  root  of 
6084. 

1)5.  REMARK. — The  operations  in  the  last  example  have 
been  performed  on  but  two  periods,  but  it  is  plain  that  the 
same  reasoning  is  equally  applicable  to  larger  numbers,  for 
by  changing  the  order  of  the  units,  we  do  not  change  the 
relation  in  which  they  stand  to  each  other. 

Thus,  in  the  number  60  84  95,  the  two  periods  60  84 
have  the  same  relation  to  each  other  as  in  the  numbtr 
60  84  ;  and  hence  the  methods  used  in  the  last  example 
are  equally  applicable  to  larger  numbers. 

96.  Hence,  for  the  extraction  of  the  square  root  of 
numbers,  we  have  the  following 

RULE. 

I.  Separate  the  given  number  into  periods  of  two  figures 
tach,  beginnijig  at  the  right  hand : — the  period  on  the  left  will 
often  contain  but  one  figure. 

II.  Find  the  greatest  square  in  the  first  period  on  the  lej>\ 
and  place  its  root  on  the  right,  after  the  manner  of  a  quotient 


QUEST.— -95.    Will  the  reasoning  in  .he  example  apply  to  more  than 
two  period*  7 

13 


138  ELEMENTARY  ALGEBRA. 

in  division.  Subtract  the  square  of  the  root  from  the  firsi 
period,  and  to  the  remainder  bring  down  the  second  period  for 
a  dividend. 

III.  Double  the  root  already  found,  and  place  it  on  the  lift 
for  a  divisor.      Seek  how  many  times  the  divisor  is  contained 
in  the  dividend,  exclusive  of  the  right-hand  figure,  and  place 
the  figure  in  the  root  and  also  at  the  right  of  the  divisor. 

IV.  Multiply  the  divisor,  thus  augmented,  by  the  last  figure 
of  the  root,  and  subtract  the  product  from  tfte  dividend,  and  to 
the  remainder  bring  down  the  next  period  for  a  new  dividend 
But  if  any  of  the  products  should  be  greater  than  the  dividend, 
diminish  the  last  figure  of  the  root. 

V.  Double  the  whole  root  already  found,  for  a  new  divisor, 
and  continue  the  operation  as  before,  until  all  the  periods  are 
*-t  ought  down. 

97.  1st  REMARK.  If,  after  all  the  periods  are  brought 
down,  there  is  no  remainder,  the  proposed  number  is  a  per- 
fect square.  But  if  there  is  a  remainder,  you  have  only 
found  the  root  of  the  greatest  perfect  square  contained  in 
the  given  number,  or  the  entire  part  of  the  root  sought. 

For  example,  if  it  were  required  to  extract  the  square 
root  of  665,  we  should  find  25  for  the  entire  part  of  the 
root,  and  a  remainder  of  40,  which  shows  that  665  is  not 
a  perfect  square.  But  is  the  square  of  25  the  greatest  per- 
fect square  contained  in  665  ?  that  is,  is  25  the  entire  part 
of  the  root  ?  To  prove  this,  we  will  first  show  that,  the 
difference  between  the  squares  of  two  consecutive  numbers,  is 
equal  to  twice  the  less  number  augmented  by  unity. 


QUEST. — 96.  Give  the  rule  for  extracting  the  square  root  of  numbers 
What  is  the  first  step  ?  What  the  second  ?  What  the  third  1  Wha 
the  fourth  !  What  the  fifth  ? 


EXTRACTION  OF  THE  SQUARE  ROOT.       139 

Let  .  .  a=    the  less  number, 

and  .  .  a-j-1    =    the  greater. 

Then  .  (a+l)2=a?+ 2a  +  l, 

and  .  .        (a)2 =a2. 

Their  difference  is       =^        2a-\-l     as  enunciated. 

Hence,  the  entire  part  of  the  root  cannot  be  augmented, 
unless  the  remainder  exceeds  twice  the  root  found,  plus 
unity. 

But  25x2+l=51>40  the  remainder:  therefore,  25  is 
the  entire  part  of  the  root. 

98.  2nd  REMARK. — The  number  of  figures  in  the  root 
will  always  be  equal  to  the  number  of  periods  into  which 
the  given  number  is  separated. 

EXAMPLES. 

1.  To  find  the  square  root  of  7225.  Ans.  85. 

2.  To  find  the  square  root  of  17689.  Ans.   133. 

3.  To  find  the  square  root  of  994009.  Ans.  997. 

4.  To  find  the  square  root  of  85673536.  Ans.  9256. 

5.  To  find  the  square  root  of  67798756.  Ans.  8234. 

6.  To  find  the  square  root  of  978121.  Ans.  989. 

7.  To  find  the  square  root  of  956484.  Ans.  978. 

8.  What  is  the  square  root  of  3^372961  ?  Ans.  6031. 

9.  What  is  the  square  root  of  22071204  1  Ans.  4698. 

10.  What  is  the  square  root  of  106929?  Ans.  327. 

11.  What  is  the  square  root  of  12088868379025  ? 

Ans.  3476905. 


QUEST. — 98.  How  many  figures  will  you  always  find  in  the  root] 


140 


ELEMENTARY  ALGEBRA. 


99.  3rd  REMARK. — If  the  given  number  has  not  an  exact 
root,  there  will  be  a  remainder  after  all  the  periods  are 
brought  down,  in  which  case  ciphers  may  be  annexed, 
forming  new  periods,  each  of  which  will  give  one  decimal 
place  in  the  root. 

1.  What  is  the  square  root  of  36729  ? 


In  this  example  there  are 
two  periods  of  decimals, 
which  give  two  places  of 
decimals  in  the  root. 


36729 
1 


191,64  + 


2  91267 
|261 


38  1 


3826 


629 
381 

24800 
22956 


3832  4 


184400 
153296 

31104  Rem. 


2.  What  is  the  square  root  of  2268741  ? 

3.  What  is  the  square  root  of  7596796  ? 

4.  What  is  the  square  root  of  96  ? 

5.  What  is  the  square  root  of  153  ? 

6.  What  is  the  square  root  of  101  ? 


Ans.  1506,23  +  . 

\ 

Ans.  2756,22  +  . 

Ans.  9,79795  +  . 
Ans.  12,36931  +  . 
Ans.  10,04987  +  . 


QUEST. — 99.  How  will  you  find  the  decimal  part  of  the  root 


EXTRACTION  OF  THE  SQUARE  ROOT.       141 

7.  What  is  the  square  root  of  285970396644  ? 

Ans.  534762. 

8.  What  is  the  square  root  of  41605800625  ? 

Ans.  203975. 

9.  What  is  the  square  root  of  48303584206084  ? 

Ans.  6950078. 

Extraction  of  the  square  root  of  Fractions. 

1  OO.  Since  the  square  or  second  power  of  a  fraction  is 
obtained  by  squaring  the  numerator  and  denominator  sepa- 
rately, it  follows  that  the  square  root  of  a  fraction  will  be 
equal  to  the  square  root  of  the  numerator  divided  by  the 
square  root  of  the  denominator. 

For  example,  the  square  root  of  — -  is  equal  to   -=- :   for 


1.  What  is  the  square  root  of    —  ?  Ans.  — . 

9  3 

2.  What  is  the  square  root  of    —  ?  Ans.  ~. 

10  4 

3.  What  is  the  square  root  of    —  ?  Ans.  — . 

4.  What  is  the  square  root  of  ?  Ans.  — . 

1  a  •* 

5.  What  is  the  square  root  of    — -  ?  Ans.  — . 

o4  2 


QUEST. — 10O.  If  the  numerator  and  denominator  of  a  fraction  are 
perfect  squares,  how  will  you  extract  the  square  root  * 
13* 


142  ELEMENTARY  ALGEBRA. 

'    6.  What  is  the  square  root  of     ^—7:7^   ?         Arts.    •—  ~ 

' 


582169  „  763 

7.  What  is  the  square  root  of     -  ?          Ans.    —  -  — 

956484  978 

1O1.  If  neither  the  numerator  nor  the  denominator  is  a 
perfect  square,  the  root  of  the  fraction  cannot  be  exactly 
found.  We  can,  however,  easily  find  the  approximate  root. 
For  this  purpose, 

Multiply  both  terms  of  the  fraction  by  the  denominator, 
which  makes  the  denominator  a  perfect  square  without  altering 
the  value  of  the  fraction.  Then,  extract  t,he  square  root  of 
the  numerator,  and  divide  this  root  by  the  root  of  the  denomi- 
tor  ;  this  quotient  will  be  the  approximate  root. 

3 

Thus,  if  it  be  required  to  extract  the  square  root  of   —  , 

0 

we  multiply,  both  terms  by  5,  which  gives    —  . 
v  25 

We  then  have 

Vr15  =  3,8729-f-  : 
hence,  3,8729-f  -r-  5  =  ,7745+  =  Ans. 

7 

2.  What  is  the  square  root  of    —  ?        Ans.  1,32287+. 

14 

3.  What  is  the  square  root  of    —  ?        Ans.  1,24721+. 

y 


4.  What  is  the  square  root  of     11—  ? 

Ib 


Ans.  3,41869  + 


QUEST. — 101.  If  the  numerator  and  denominator  of  a  fraction  are  not 
perfect  squares,  how  do  you  extract  the  square  root ! 


EXTRACTION  OF  THE  SQUARE  ROOT.       143 

5.  What  is  the  square  root  of     ?i|  1      Ans.  2,7131  3  -f-. 

6.  What  is  the  square  root  of     8  —  ?      Ans.  2,88203  +  . 

49 

7.  W^at  is  the  square  root  of    —  ?        Ans.  0,64549  f. 

12 


8.  What  is  the  square  root  of     10—  1 


Ans.  3,20936  +  . 


1O2.  Finally,  instead  of  the  last  method,  we  may,  if  we 
please, 

Change  the  vulgar  fraction  into  a  decimal,  and  continue  the 
division  until  the  number  of  decimal  places  is  double  the  number 
of  places  required  in  the  root.  Then,  extract  the  root  of  the 
decimal  by  the  last  rule. 

Ex.  1.  Extract  the  square  root  of    —     to  within    ,001. 

This  number,  reduced  to  decimals,  is   0.785714  to  within 
0,000001  ;  but  the  root  of  0,785714  to  the  nearest  unit,  is 

,886  ;  hence  0,886  is  the  root  of    —     to  within  ,001. 

14 

2.  Find  the    V/2—     to  within  0,0001. 

Ans.  1,6931  + 

3.  What  is  the  square  root  of    —  ?        Ans.  0,24253+  , 

4.  What  is  the  square  root  of    —  -  %        Ans.  0,93541+. 

o 

5.  What  is  the  square  root  of    —  ?        Ans.  1,29099+. 

o 


QUEST.— 102.  By  what  other  method  may  the  root  be  found  1 


144  ELEMENTARY  ALGEBRA. 


Extraction  of  the  Square  Root  of  Monomials. 

1O3.  In  order  to  discover  the  process  for  extracting  the 
square  root,  we  must  see  how  the  square  of  the  monomia. 
is  formed. 

By  the  rule  for  the  multiplication  of  monomials  (Art.  35), 
we  have 


that  is,  in  order  to  square  a  monomial,  it  is  necessary  to 
square  its  coefficient,  and  double  each  of  the  exponents  of  the 
different  letters.  Hence,  to  find  the  root  of  the  square  of  a 
monomial,  we  have  the  following 

RULE. 

I.  Extract  the  square  root  of  the  coefficient. 

II.  Divide  the  exponent  of  each  letter  by  2. 

Thus,     -y^G4a6Z>4  =.  8a3b2     for 


2.  Find  the  square  root  of   625a268c6.  Ans.  25ab*c*. 

3.  Find  the  square  root  of  576a4bGc8.          Ans.  24a2b3c*. 

4.  Find  the  square  root  of   196x6y2z4.          Ans.  I4x3yzz. 

5.  Find  the  square  root  of  44la8beclod16. 

Ans. 

6.  Find  the  square  root  of  784al2bucl6d2. 

Ans. 

7.  Find  the  square  root  of  81a8&4c6. 

Ans. 


QDEST. — 103.  How  do  you  extract  the  square  root  of  a  monomial 


EXTRACTION  OF  THE  SQUARE  ROOT.      145 

1O4.  From  the  preceding  rule  it  follows,  that  when  a 
monomial  is  a  perfect  square,  its  numerical  coefficient  is  a 
perfect  square,  and  all  its  exponents  even  numbers.  Thus, 
25o4£2  is  a  perfect  square,  but  98a64  is  not  a  perfect  square, 
because  98  is  not  a  perfect  square,  and  a  is  affected  with 
an  uneven  exponent. 

In  the  latter  case,  the  quantity  is  introduced  into  the  cal- 
culus by  affecting  it  with  the  sign  -y/  ,  and  it  is  written 
thus  : 


Quantities  of  this  kind  are  called  radical  quantities,  or  irra- 
tional quantities,  or  simply  radicals  of  the  second  degree. 
They  are  also,  sometimes  called  Surds. 

These  expressions  may  often  be  simplified,  upon  the  prin- 
ciple that,  the  square  root  of  the  product  of  two  or  more  factors 
is  equal  to  the  product  of  the  square  roots  of  these  factors  ;  or, 
in  algebraic  language, 

*\/abcd  .  .  .   =-y/a  .  •y'  b  .  -y/c  .  -y/  d  .  .  . 

This  being  the  case,  the  above  expression, 
be  put  under  the  form 


Now  V49M  may  be  reduced  to  7bz  ;  hence, 


In  like  manner, 


146  ELEMENTARY  ALGEBRA. 

The  quantity  which  stands  without  the  radical  sign  is 
called  the  coefficient  of  the  radical.     Thus,  in  the  expressions 


the  quantities  7£2,  3abc,  12a£2c5,  are  called  coefficients  of 
the  radicals. 

Hence,  to  simplify  a  radical  expression  of  the  second 
degree,  we  have  the  following 


RULE. 

I.  Separate  the  expression  into  two  parts,  of  which  one  shall 
contain  all  the  factors  that  are  perfect  squares,  and  the  other 
the  remaining  ones. 

II.  Take  the  roots  of  the  perfect  squares  and  place  them 
before  the  radical  sign,  under  which  leave  those  factors  which 
are  not  perfect  squares. 

1O5.  REMARK. — To  determine  if  a  given  number  has 
any  factor  which  is  a  perfect  square,  we  examine  and  see 
if  it  is  divisible  by  either  of  the  perfect  squares 

4,     9,     16,     25,     36,     49,     64,     81,  &c ; 

and  if  it  is  not,  we  conclude  that  it  does  not  contain  a  fac- 
tor which  is  a  perfect  square. 


QUEST. — 104.  When  is  a  monomial  a  perfect  square  7  When  it  is 
not  a  perfect  square,  how  is  it  introduced  into  the  calculus  1  What  are 
quantities  of  this  kind  called  7  May  they  be  simplified  !  Upon  whal 
principle  1  What  is  a  coefficient  of  a  radical  1  Give  the  rule  for  reducing 
radicals. — 105.  How  do  you  determine  whether  a  given  number  has  a 
factor  which  is  a  perfect  square  1 


EXTRACTION    OF    THE    SQUARE    ROOT.  147 


EXAMPLES. 

1.  Reduce      -\/75d3bc     to  its  simplest  form. 

Ans. 

2.  Reduce      yTSsFo^     to  its  simplest  form. 

Ans. 

3.  Reduce      -\/32a9b8c     to  its  simplest  form 

Ans. 

4.  Reduce      -\/256a2b*c8     to  its  simplest  form. 

Ans. 

5.  Reduce      Vl024a967e*     to  its  simplest  form. 

Ans. 


6.  Reduce      -v/729a765c6J    to  its  simplest  form. 


7.  Reduce 


'ft  | 

,/      I 

:  s       "|/!ri"|lj 
4l<!l 


to  its  simplest  form, 
^n^. 


i  ^ts  simplest  form. 


tii 


its  simplest  form 

An.y.  ISO4*/3* 

its  simplest  form 

Ans. 
s  simplest  form. 
Ans. 


148 


ELEMENTARY  ALGEBRA. 


1O6.  Since  like  signs  in  both  the  factors  give  a  plus 
sign  in  the  product,  the  square  of  —  a,  as  well  as  that  of 
-\-a,  will  be  a2;  hence  the  root  of  a2  is  either  -{-a  or  —a 
Also,  the  square  root  of  25a?b4  is  either  -\-5ab2  or  —-dab2. 
Whence  we  may  conclude,  that  if  a  monomial  is  positive 
its  square  root  may  be  affected  either  with  the  sign  -f  or 
—  ;  thus,  V9«4=dc3a2  ;  for,  +  3a2  or  —  3a2,  squared, 
gives  9a4.  The  double  sign  ±  with  which  the  root  is 
affected  is  read  plus  or  minus. 

If  the  proposed  monomial  were  negative,  it  would  be  im- 
possible to  extract  its  root,  since  it  has  just  been  shown  that 
the  square  of  every  quantity,  whether  positive  or  negative, 
is  essentially  positive.  Therefore, 


are  algebraic  symbols  which  indicate  operations  that  cannot 
be  performed.  They  are  called  imaginary  quantities,  or 
rather  imaginary  expressions,  and  are  frequently  met  with 
in  the  resolution  of  equations  of  the  second  degree.  These 
symbols  can,  however,  by  extending  the  rules,  be  simplified 
in  the  same  manner  as  those  irrational  expressions  which 
indicate  operations  that  cannot  be  performed.  Thus,  V—  9 
may  be  reduced  by  (Art.  1O4).  Thus, 


also, 


QUEST. — 106.  What  sign  is  placed  before  the  square  root  of  a  mono- 
mial 1  Why  may  you  place  the  sign  plus  or  minus'?  What  is  an  ima- 
ginary quantity  1  Why  is  it  called  imaginary  1 


RADICALS    OF    THE    SECOND    DEGREE.  149 


Of  the  Calculus  of  Radicals  of  the  Second  Degree 

107.  A  radical  quantity  is  the  indicated  root  of   an 
imperfect  power. 

The  extraction  of  the  square  root  gives  rise  to  such  ex- 
pressions as  yX  3  -\/b,  7  -\/2,  which  are  called  irra- 
tional quantities,  or  radicals  of  the  second  degree.  We  will 
now  establish  rules  for  performing  the  four  fundamental 
operations  on  these  expressions. 

108.  Two  radicals  of  the  second  degree  are  similar^ 
when  the  quantities  under  the-  radical  sign  are  the  same  in 
both.     Thus,    3  \/b    and    5e  yT~~  are  similar  radicals;  and 
BO  also  are    9  ^/2    and    7  -v/57 


Addition. 

1O9.    Radicals   of  the   second  degree  may  be  added 
together  by  the  following 

RULE. 

I.  If  the  radicals  are  similar  add  their  coefficients,  and  to 
the  sum  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar,  connect  them  together 
with  their  proper  signs. 

Thus, 


QUEST. — 107.  What  is  a  radical  quantity  1  What  are  such  quantities 
called? — 1-08.  When  are  radicals  of  the  second  degree  similar  1 — 
109.  How  do  you  add  similar  radicals  of  the  second  degree  *  How  do 
vou  add  radicals  which  are  not  similar  1 

14 


150  ELEMENTARY  ALGEBRA. 

In  like  manner. 


Two  radicals,  which  do  not  appear  to  be  similar  at  first 
sight,  may  become  so  by  simplification  (Art.  1O4). 
For  example, 


and 

When  the  radicals  are  not  similar,  the  addition  or  sub- 
traction can  only  be  indicated.  Thus,  in  order  to  add 
3  -\/b  to  5  -y/a,  we  write 


EXAMPLES. 


1.  What  is  the  sum  of     ^/27a2     and      -y/48a2  ? 

Ans. 

2.  What  is  the  sum  of     -/50a4£2     and      </72a*b2  ? 


3.  \Vhat  is  the  sum  of        --    and 

5  V     15 


4.  What  is  the  sum  of      Vl25     and      l/500a2  ? 


RADICALS    OF    THE    SECOND    DEGREE.  151 


/  50  /I  00   „ 

5.   VV  hat  is  the  sum  of  \  I  ———     and    \  /-—  -  —  ? 
V    147  V   294 


6.  What  is  the  sum  of    y98a?x     and      i/36x2—  36a2 


"    What  is  the  sum  of    -y/98a2ar     and 

^4^. 
S    Required  the  sum  of    -v/72"    and      -y/128. 


.  14 
9.  Required  the  sum  of    -v/27     and      -/1  47. 

Ans.  10 

27 


/2* 
10.  Required  the  sum  of  \  /  —    -^nd 


Ans.  g 
11.  Required  the  sum  of   2  y/lffi     and     3 


12.  Required  the  sum  of    y^43"    and     10-/3637 

jlns.  119/3 

13.  What  is  the  sum  of     ^/32Qa2b2     and 


14    What  is  the  sum  of     -TSo^F     and 


152  ELEMENTARY  ALGEBRA. 


Subtraction. 

1 1  O.  To  subtract  one  radical  expression  from  another, 
we  have  the  following 

RULE. 

I.  If  the  radicals  are  similar,  subtract  their  coefficients, 
and  to  the  difference  annex  the  common  radical. 

II.  If  the  radicals  are  not  similar,  their  difference  can  only 
be  indicated  by  the  minus  sign. 


EXAMPLES. 

1.  What  is  the  difference  between    3ai/~b    and    a 

Here'   ^a^/^b—a-^T)=2a^/~JT  Ans. 

2.  From    9ay27F  subtract    6a  V  27bz. 
First,        9ay§7P"=:27«&'/~37  and    6a  ^/l27b* 
and  27abi~3'—18ab<~3  =  9abi~3    Ans. 


3.  What  is  the  difference  of    \f~75    and     -\/~48  1 

Ans. 


4.  What  is  the  difference  of    ^24aW  and    V  54&4  I 

Ans.     2ab— 


QUEST. — 110.  How  do  you  subtract  similar  radicals  1     How  do  you 
subtract  radicals  which  are  not  similar  ? 


RADICALS    OF    THE    SECOND    DEGREE.  153 


5.  Required  the  difference  of  it-r-     and    \«=< 


40 

6.  What  is  the  difference  of    ^f\^S~aW   and     -/32«9  ? 


7.  What  is  the  difference  of    ^/48a3b3    and 

^4/i.s.  4ab 

8.  What  is  the  difference  of   V242a565    and 


/3~  /3" 

9.  What  is  the  difference  of  V/-T-     and    \/  —  ? 


10.  What  is  the  difference  of    V320«2    and 

1  1    What  is  the  difference  between 
and 


12.  What  is  the  difference  between 
and 


13.  What  is  the  difference  between 
and 

14* 


1 54  ELEMENTARY  ALGEBRA. 


Multiplication. 

111.  For  the  multiplication  of  radicals,  we  have  the 
following 

RULE. 

I.  Multiply  the  quantities  under  the  radical  signs  together, 
and  place  the  common  radical  over  the  product. 

II.  If  the  radicals  have  coefficients,  we  multiply  them  to- 
gether, and  place  the  product  before  the  common  radical. 

Thus,  yV  X  V^~=  Va5 ; 

This  is  the  principle  of  Art.    1O4,   taken  in  the  inverse 
order. 


EXAMPLES. 


1.  What  is  the  product  of   3  ^5aB   and    4 

Ans. 


2.  What  is  the  product  of   2a  ^/fic    and 

Ans.  6a?be 


3.  What  is  the  product  of  2a  ^a2+b*  and  —  3a 

Ans.   —6az(az+b2). 


QUEST. — 111.  How  do  you  multiply  quantities  which  are  under  radi- 
cal signs  1  When  the  radicals  have  coefficients,  now  do  you  multiple 
them  1 


RADICALS    OF    THE    SECOND    DEGREE.  155 

4.  What  is  the  product  of    3  -\^~  and    2  -y/ST 

Ans.  24. 


5.  What  is  the  product  of   f  -y/l^    and    T2o  V%c*b  • 

Ans. 


6.  What  is  the  product  of   2x4  ^fb    and    2x— 

Ans.  4xz—b. 

7.  What  is  the  product  of 

and 


8.  What  is  the  product  of    3a-/2703     by     -/^  ? 


Division. 

112.  To  divide  one  radical  by  another,  we  have  the 
following 

RULE. 

I.  Divide  one  of  the  quantities  under  the  radical  sign  by  the 
other,  and  place  the  common  radical  jver  the  quotient. 

II.  If  the  radicals  have  coefficients,  divide  the  coefficient  of 
the  dividend  by  the  coefficient  of  the  divisor,   and  place  the 
quotient  before  the  common  radical. 


QUEST. — 112.  How  do  you  divide  quantities  which  are  under  the 
radical  sign  1  When  the  radicals  have  coefficients,  how  do  you  divide 
them? 


156  ELEMENTARY  ALGEBRA. 

Thus,     — — =\/—  ;     for  the    squares  of  these  two 

expressions  are  equal  to  the  same  quantity     —  ;     hence 
the  expressions  themselves  must  be  equal. 


EXAMPLES. 


1.  Divide     5«yT    by     2&  V^-  Ans-  ^T 


2.  Divide  Wac^Wc     by  4ci/2F.  Ans. 

3.  Divide  6a-v/96F    by  3^8P.  Ans.  4abi/3. 

4.  Divide  4a2-/50F    by  2a2-y/5£7  Ans. 

5.  Divide  26a?b^/8laW  by     1 


6.  Divide     84ff3^4  ^27  ac     by 


7.  Divide  -y/F2     by     ^2- 

8.  Divide  Ga^y^a3     by     12-/5a.  Ans. 

9.  Divide  Ga-v/ToF"  by     3-/5T  A 

10.  Divide  48b^^/l5'   by     2i2VS  ^^-  36062. 

11.  Divide  Sa^c3-/^     by     20^2837 


2ab*c3d. 
12.  Divide     96a4c3-v/98F    by 


RADICALS    OF    THE    SECOND    DEGREE.  157 

13.  Divide     27a556  i/2Ttf     by     ^/^a~. 

Ans. 

14.  Divide     ISaW-^&T*     by     &ab^/tf~ 

Ans. 

To  Extract  the  Square  Root  of  a  Polynomial. 

113.  Before  explaining  the  rule  for  the  extraction  of 
the  square  root  of  a  polynomial,  let  us  first  examine  the 
squares  of  several  polynomials  :  we  have 


The  law  by  which  these  squares  are  formed  can  be  enun- 
ciated thus  : 

The,  square  of  any  polynomial  contains  the  square  of  the 
first  term,  plus  twice  the  product  of  the  first  term,  by  the  second, 
plus  the  square  of  the  second  ;  plus  twice  the  first  two  terms 
multiplied  by  the  third,  plus  the  square  of  the  third  ;  plus  twice 
the  first  three  terms  multiplied  by  the  fourth,  plus  the  square 
of  the  fourth;  and  so  on. 


QUEST. — 113.  What  is  the  square  of  a  binomial  eqaalto?  What 
is  the  square  of  a  trinomial  equal  to  ?  What  is  the  square  of  any 
polynomial  equal  to  ? 


1 58  ELEMENTARY  ALGEBRA. 

114.  Hence,  to  extract  the  square  root  of  a  polynomial 
we  have  the  following 

RULE. 

I.  Arrange  the  polynom ' zl  with  reference  to  one  of  its  letters 
and  extract  the  square  root  of  the  first  term  :  this  will  give  the 
first  term  of  the  root. 

II.  Divide  the  second  term  of  the  polynomial  by  double  the 
first  term  of  the  root,  and  lite  quotient  will  be  the  second  term 
of  the  root. 

III.  Then  form  the  square  of  the  two  terms  of  the  root 
found,  and   subtract  it  fi  jm  the  first  polynomial,  and  then 

divide  the  first  term  of  the  remainder  by  clouble  the  first  term 
of  the  root,  and  the  quotient  will  be  the  third  term. 

IV.  Form  the  double  products  of  the  first  and  second  terms, 
by  the  third,  plus  the  square  of  the  third ;  then  subtract  all 
these  products  from  the  last  remainder,  and  divide  the  first 
term  of  the  result  by  double  the  first  term  of  the  root,  and  the 
quotient  will  be  the  fourth  term.      Then  proceed  in  the  sanw 
manner  to  find  the  other  te*ms. 

EXAMPLES. 

X     Extract  the  square  loot  of  the  polynomial 


First  arrange  it  with  reference  to  the  letter  a. 

25a4— 
25a4— 


10a2 
40a2&2-^4tfZ>3-|-16&4     1st  Rem. 

2d  Rem. 


RADICALS    OF    THE    SECOND    DEGREE.  159 

After  having  arranged  the  polynomial  with  reference  to  a, 
extract  the  square  root  of  25a4,  this  gives  5a2,  which  is 
placed  at  the  right  of  the  polynomial ;  then  divide  the 
second  term,  — 30«36,  by  the  double  of  5a2,  or  10a2;'the 
quotient  is  —  3ab,  and  is  placed  at  the  right  of  5a2.  Hence, 
the  first  two  terms  of  the  root  are  5a2— Sab.  Squaring  this 
binomial,  it  becomes  25a4  — 30«3i-|-9a2&2,  which,  subtracted 
from  the  proposed  polynomial,  gives  a  remainder,  of  which 
the  first  term  is  40a262.  Dividing  this  first  term  by  10<z2, 
(the  double  of  5a2),  the  quotient  is  -f-4&2;  this  is  the  third 
term  of  the  root,  and  is  written  on  the  right  of  the  first  two 
terms.  By  forming  the  double  product  of  5a2  —  3ab  by  4&2, 
and  at  the  same  time  squaring  4&2,  we  find  the  polynomial 
4Qa2b2—24ab3+l6b*,  which,  subtracted  from  the  first  re- 
mainder, gives  0.  Therefore  5a2  —  3ab  +  4b2  is  the  required 
root. 

2.  Find  the  square  root  of      a*-{-4a3x-\-6a2x2-r4ax3-\-x4. 

Ans.  at  +  Zax+x2. 

3.  Find  the  square  root  of      a*~-4a3x-\-6a?x2— 4ax3-[-x*. 

Ans.  a2—2ax-{-x2. 

4.  Find  the  square  root  of 

4#6+  12a;5+5*4— 2x3+7x2— 2a?+ 1 . 
Ans. 

5.  Find  the  square  root  of 

9a4  —  12«36-f28a262 

Ans.  3a2— 


QUEST. — 114.  Give  the  rule  for  extracting  the  square  root  of  a  poly- 
nomial 1  What  is  the  first  step  1  What  the  second  1  What  the  third  ' 
What  the  fourth  1 


1  60  ELEMENTARY  ALGEBRA. 

G.  What  is  the  square  root  of 

a;4  —  4ax3  +  4a2x2  —  4x2  -f  Sax  -f  4. 

Ans.  x2—2ax—2. 
7.  What  is  the  square  root  of 


Ans.  3a?+y—  2. 
8.  What  is  the  square  root  of    y4—  2y2x2  +  2x2—  2y2+l 


9.  What  is  the  square  root,  of     9a464  —  30<z3£3+25a2£2? 

Arcs.  3a2&2  —  5a 

10.  Find  the  square  root  of 

25a*b2  —  40«362c  -f  76a252c2  —  48abzc3  +  3  6i 
—  36«2^c3  +  9«4c2. 
Ans.  5a2b  —  3a2c  — 


115.  We  will  conclude  this  subject  with  the  following 
remarks. 

1st.  A  binomial  can  never  be  a  perfect  square,  since  we 
know  that  the  square  of  the  most  simple  polynomial,  viz  : 
a  binomial,  contains  three  distinct  parts,  which  cannot  ex- 
perience any  reduction  amongst  themselves.  Thus,  the 
expression  a2-\-b2  is  not  a  perfect  square  ;  it  wants  the  term 
±2ab  in  order  that  it  should  be  the  square  of  a±b. 

2nd.  In  order  that  a  trinomial,  when  arranged,  may  be  a 
perfect  square,  its  two  extreme  terms  must  be  squares,  and 
the  middle  term  must  be  the  double  product  of  the  square 
roots  of  the  two  others.  Therefore,  to  obtain  the  square 
root  of  a  trinomial  when  it  is  a  perfect  square  ;  Extract  the 
roots  of  the  two  extreme  terms,  and  give  these  roots  the  same 
or  contrary  signs,  nrcnrding  as  the  middle  term  is  positive  or 


RADICALS    OF    THE    SECOND    DEGREE. 


161 


negative.      To  verify  «£,  see  if  the  double  product  of  the  two 
roots  gives  the  middle  to*-m  of  the  trinomial.     Thus, 

Qa6— 4Qa4:02 4-G4«2&4     is  a  perfect  square, 

since  -y/906  =  3a3,  and    <v/64a2&4=  —  8abz, 

and  also     2  x  3a3  X  —  8ab2=  —  48«452=  the  middle  term. 

But  4az-\-l4ab+9b2  is  not  a  perfect  square  :  for  although 
4a2  and  -I-  9b2  are  the  squares  of  20  and  3£,  yet  2  X  2a  X  3i 
is  not  equal  to  I4ab. 

3rd.  In  the  series  of  operations  required  in  a  general  ex- 
ample, when  the  first  term  of  one  of  the  remainders  is  not 
exactly  divisible  by  twice  the  first  term  of  the  root,  we  may 
conclude  that  the  proposed  polynomial  is  not  a  perfect 
square.  This  is  an  evident  consequence  of  the  course  of 
reasoning,  by  which  we  have  arrived  at  the  general  rule  for 
extracting  the  square  root. 

4th.  When  the  polynomial  is  not  a  perfect  square,  it  may 
be  simplified  (See  Art.  1O4.) 


Take,  for  example,  the  expression     \/a3b  +  4a2b2 -\-4ab3. 

The  quantity  under  the  radical  is  not  a  perfect  square ; 
but  it  can  be  put  under  the  form  ab(a2-\-4ab-{-4:b2).  Now, 
the  factor  between  the  parenthesis  is  evidently  the  square 
of  a+26,  whence  we  may  conclude  that, 


2.  Reduce  V2a26 — 4ab2  -f  2b3    to  its  simple  form 

Ans.  (a  — 


QUEST. — 115.  Can  a  binomial  ever  be  a  pertect  power!  Why  not  1 
When  is  a  trinomial  a  perfect  square  1  When,  in  extracting  the  square 
root  we  find  that  the  first  term  of  the  remainder  is  not  divisible  by  twice  the 
root,  in  the  polynomial  a  perfect,  power  or  not 7 

15 


162  KLKiMENTARY    ALGEBRA. 


CHAPTER  VI. 
Equations  of  the  Second  Degree. 

116*  An  Equation  of  the  second  degree  is  c.ie  in  which 
the  greatest  exponent  of  the  unknown  quantity  is  equal  to  2. 

If  the  equation  contains  two  unknown  quantities,  it  is  of 
the  second  degree  when  the  greatest  sum  of  the  exponents 
with  which  the  unknown  quantity  is  affected,  in  any  term,  is 
equal  to  2.  Thus,  x 

a?=a,     ax2-\-bx=c,    and     xy  +  x=d*, 
are  equations  of  the  second  degree. 

117.  Equations  of  the  second  degree  are  divided  into  two 
classes : 

1st.  Equations  which  involve  only  the  square  of  the  un- 
known quantity  and  known  terms.  These  are  called,  Incom- 
plete Equations. 

2d.  Equations  which  involve  the  first  and  second  powers 
of  the  unknown  quantity,  aad  known  terms.  These  are 
called,  Complete  Equations. 

QUEST.— 1 16.  What  is  an  equation  of  the  second  degree  ? — 1 1 7.  Into 
how  many  classes  are  equations  of  the  second  degree  divided?  What 
i«  an  incomplete  equation  ?  What  is  a  complete  equation  ? 


EQUATIONS  OF   THE   SECOND   DEGREE.  163 


Thus,  a^+S*2  -5=7 

and  5s2—  3a*—  4=a 

are  incomplete  equations  :  and 

3x*-5ar—  3.r2  +  a=6 
Sx2—  Sx2-  X—  c=d 

are  complete  equations. 

O/*  Incomplete  Equations. 

118.  If  we  take  an  incomplete  equation  of  the  form 

14x2-8x2=40-2xil 
we  have,  by  collecting  the  coefficients  of  x2, 

8x2=40,  or  ^=5. 
Again,  —  if  we  have  the  equation 

ax2  +  fcx2  +  </=/, 
we  shall  have, 

==W     and    ar2==m 


by  substituting  m  for  the  known  terms  which  compose  the 
second  member.     Hence, 

Every  incomplete  equation  can  be  reduced  to  an  equation 
involving  two  terms,  of  the  form 
ar2=m, 

and  from  this  circumstance  the  incomplete  equations  are  often 
called  equations  involving  two  terms. 

From  which  we  have,  by  extracting  the  square  root  of  both 
members,  x 


QUEST.—  118,  To  what  form  may  every  incomplete  equation  be 
reduced  ?    What  are  incomplete  equations  often  called  ? 


164  ELEMENTARY   ALGEBRA. 

1.  What  number  is  that  which  being  multiplied  by  itself 
the  product  will  be  144. 

Let  x=  the  number:  then 

x  Xx=  a?=144. 

It  is  plain  that  the  value  of  x  will  be  found  by  extracting 
the  square  root  of  both  members  of  the  equation  :  that  is 


2.  A  person  being  asked  how  much  money  he  had,  said 
if  the  number  of  dollars  be  squared  and  6  be  added,  the  sum 
will  be  42  :  How  much  had  he  ? 

Let  x=  the  number  of  dollars. 
Then  by  the  conditions 

*2+6=42: 

hence,  a*=42-6=36 

and  #=6. 

Ans.  $6 

3.  A  grocer  being  asked  how  much  sugar  he  had  sold  to  a 
person,  answered,  if  the  square  of  the  number  of  pounds  be 
multiplied  by  7,  the  product  will  be  1575.   How  many  pounds 
had  he  sold  ? 

Denote  the  number  of  pounds  by  x. 
Then  by  the  conditions  of  the  question 

7^=1575: 

ience,  s*=225 

and  x  =15. 

Jlns.  15 


EQUATIONS    OP    THE    SECOND    DEGREE.  165 

4.  A  person  being  asked  his  age  said,  if  from  the  square 
of  my  age  you  take  192,  the  remainder  will  be  the  square 
of  half  my  age  :  what  was  his  age  ? 

Denote  his  age  by  x. 

Then,  by  the  conditions  of  the  question 


and  by  clearing  the  fractions 


hence,  4x2—  xz=768, 

and  3a?2=768 

*2=256 
x  =   16. 


Ans.  16. 


5.  What  number  is  that  whose  eighth  part  multiplied  by 
its  fifth  part  and  the  product  divided  by  4,  shall  give  a  quotient 
equal  to  40  ? 

Let  x=  the  number, 

By  the  conditions  of  the  question 


hence,  _g.=40 

by  clearing  fractions, 

*a=6400 
x=  80. 

Ans.  80. 
15* 


166  ELEMENTARY   ALGEBRA. 

119.  Hence,  to   find   the  value  of  x  we  have  the  fol- 
lowing 

RULE. 

1.  Find  the  value  of  x2 ;  and  then  extract  the  square  root 
of  both  members  of  the  equation. 

4.  What  is  the  value  of  x  in  the  equation 

3x*-\-8  =  5x2  — 10. 

By  transposition     3x2 — 5x2  =  —  1 0 — 8, 
by  reducing  —  2x2=  — 18, 

by  dividing  by  2  and  changing  the  signs 

a:2  =  9, 
by  extracting  the  square  root     x  =  3. 

We  should,  however,  remark  that  the  square  root  of  9, 
is  either     +3,     or     —3.     For, 

-f3x+3  =  9     and     — 3x— 3  =  9. 
Hence,  when  we  have  the  equation 

^=9, 
we  have  x  =4-3     and     x  =—3. 

1 2O.  A  root  of  an  equation  is  any  expression  which  being 
substituted  for  the  unknown  quantity,  will  satisfy  the  equa- 
tion, that  is,  render  the  two  members  equal  to  each  other. 
Thus,  in  the  equation 

rr2  =  9 

there  are  two  roots,    -f3    and    —3  ;     for  either  of  these 
numbers  being  substituted  for  x  will  satisfy  the  equation. 


EQUATIONS  OF   THE   SECOND   DEGREE.  167 

7.  Again,  if  we  take  the  equation 

x*=m, 
we  shall  have 

x=+/m    and     ar=  —  /nT 


For, 
and 

Hence  we  may  conclude, 

1st.  That  every  incomplete  equation  of  the  second  degree 
has  two  roots. 

2d.  That  these  roots  are  numerically  equal,  but  have  con- 
trary signs. 

8.  What  are  the  roots  of  the  equation 


.  a?=+4  and  x=—  4. 
9.  What  are  the  roots  of  the  equation 


dns.  #=-f9  and  x—  —  9. 

10.  What  are  the  roots  of  the  equation 

4^+13-2^=45. 

Ans.  x=+4  and  x=  —  4 

QUEST.  —  1  19.  How  do  you  resolve  an  incomplete  equation  1  120.  What 
is  the  root  of  an  equation?  What  are  the  roots  of  the  equation  x2  =9? 
Of  the  equation  z2—  m?  How  many  roots  has  every  incomplete  equa- 
tion ?  How  do  those  roots  compare  with  each  other  ? 


1  68  ELEMENTARY  ALGEBRA. 

8.  What  are  the  roots  of  the  equation 


Ans.  *=4-2,     ar=Lj—  2 
9.  What  are  the  roots  of  the  equation. 


Ans.  #=-f-5,     x=—  5 


1  0.  Find  a  number  such  that  one-third  of  it  multiplied 
by  one-fourth  shall  be  equal  to  108  ? 

Ans.  36. 

1  1  .  What  number  is  that  whose  sixth  part  multiplied  by 
its  fifth  part  and  product  divided  by  ten,  shall  give  a  quo- 
tient equal  to  3  ? 
%  Ans.  30. 

12.  What  number  is  that  whose  square,  plus  18,  shall  be 
equal  to  half  its  square  plus  30J. 

Ans.  5. 

13.  What  numbers  are  those  which.  are  to  each  other  as 
1  to  2  and  the  difference  of  whose  squares  is  equal  to  75. 

Let        a?=     the  less  number. 

Then  2x=     the  greater. 

Then  by  the  conditions  of  the  question 


hence,  3a?2=75  ; 

and  by  dividing  by     3,    *2=25     and     at  =5, 

and  2op=10. 

Ans.  5  and    10. 


EQUATIONS    OF    THE    SECOND    DEGREE.  169 

14.  What  two  numbers  are  those  which  are  to  each  other 
as  5  to  6,  and  the  difference  of  whose  squares  is  44. 

Let  x=    the  greatest  number. 

^ 

Then   —-#=    the  least. 
6 

By  the  conditions  of  the  question 

*'-H^=44. 
36 

by  clearing  fractions, 

36JT2— 25ff2=1584; 

hence,  1 10*= 1584, 

and  a*=144, 

hence,  x  =12, 


and  -       =10- 

o 


Ans.  10  and  12. 


15.  What   two  numbers  are   those  which  are  to  each 
other  as  3  to  4,  and  the  difference  of  whose  squares  is  28  ? 

Ans.  6  and  8. 

16.  What  two  numbers  are  those  which  are  to  each  other 
as  5  to  11,  and  the  sum  of  whose  square  is  584  ? 

Ans.  10  and  22. 

17.  A  says  to  B,  my  son's  age  is  one  quarter  of  yours, 
and  the  difference  between  the  squares  of  the  numbers  re- 
presenting their  ages  is  240  :  what  were  their  ages  ? 


Younger    4. 


170  ELEMENTARY  ALGEBRA. 


When  there  are  two  unknown  quantities. 

121.  When  there  are  two  or  more  unknown  quantities, 
eliminate  one  of  them  by  the  rule  of  A  rticle  7  7  :  there  will 
thus  arise  a  new  equation  with  but  a  single  unknown  quantity, 
the  value  of  which  may  be  found  by  the  rule  already  given. 

1.  There  is  a  room  of  such  dimensions,  that  the  differ- 
rence  of  the  sides  multiplied  by  the  less  is  equal  to  36,  and 
the  product  of  the  sides  is  equal  to  360  :  what  are  the 
sides  ? 

Let   a?=    the  less  side  ; 
y=.    the  greater. 

Then,  by  the  1  st  condition, 


and  by  the  2nd,  <ry=360. 

From  the  first  equation,  we  have 

and  by  subtraction,  ar2=:324. 

Hence,  x=^324~=18; 

_  J60  _ 

Ans.x—\Qty  —  '20. 

QUEST.-  -121.  How  do  you  resolve  the  equation  when  there  are  two 
or  more  unknown  quantities  1 


EQUATIONS  OF  THE  SECOND  DEGREE.       171 

2.  A  merchant  sells  two  pieces  of  muslin,  which  together 
measure    12   yards.     He   received  for  each  piece  just  so 
many  dollars  per  yard  as  the  piece  contained  yards.     Now, 
he  gets  four  times  as  much  for  one  piece  as  for  the  other 
how  many  yards  in  each  piece  * 

Let    a:—    the  number  in  the  larger  piece  ; 
y—    the  number  in  the  shorter  piece. 

Then,  by  the  conditions  of  the  question, 


xXx=x2=  what  he  got  for  the  larger  piece  ; 
yXy=yz  =  what  he  got  for  the  shorter. 
And  a:2=i4y2,  by  the  2nd  condition. 

a;  =2y,  by  extracting  the  square  root 
Substituting  this  value  of  x   in  the  first  equation,  we  have 

y+2y=12; 
and  consequently,  y—   4, 

and  *=  8. 

AHS.  S  and  4. 

3.  What  two  numbers  are  those  whose  product  is  30,  and 
quotient  3£  1  Ans.   10  and  3. 

4.  The  product  of  two  numbers  is  a,  and  their  quotient 
•  :  what  are  the  numbers  ? 


Ans.   i/  ab  and  \l  - 


a 
~b 

5.     The  sum  of  the  squares  of  two  numbers  is  117,  an 
u*e  difference  of  their  squares  45  :    what  are  the  -numbers  ? 

Ans.  9  and  6 


172  ELEMENTARY  ALGEBRA. 

6.  The  sum  of  the  squares  of  two  numbers  is  «,  and  the 
difference  of  their  squares  is  b  :  what  are  the  numbers  ? 


7.  What  two  numbers  are  those  which  are  to  each  other 
as  3  to  4,  and  the  sum  of  whose  squares  is  225  ? 

Ans.  9  and  12. 

8.  What  two  numbers  are  those  which  are  to  each  other 
as  m  to  »,  and  the  sum  of  whose  squares  is  equal  to  a2  1 

ma  na 

Ans. 


9.  What  two  numbers  are  those  which  are  to  each  other 
as  1  to  2,  and  the  difference  of  whose  squares  is  75  ? 

Ans.  5  and  10. 

10.  What  two  numbers  are  those  which  are  to  each  other 
as  m  to  n,  and  the  difference  of  whose  squares  is  equal 
to  #>? 

mb  nb 


11.  A  certain  sum  of  money  is  placed  at  interest  for  six 
months,  at  8  per  cent,  per  annum.     Now,  if  the  amount  be 
multiplied  by  the  number  expressing  the  interest,  the  pro- 
duct will  be  562500  :  what  is  the  amount  at  interes*  ? 

Ans.   $3750. 

12.  A  person  distributes  a  sum  of  money  between  a  num- 
ber of  women  and  boys.     The  number  of  women  is  to  the 
number  of  boys  as  3  to  4.     Now,  the  boys  receive  one- 
half  as  many  dollars  as  there  are  persons,  and  the  women 
twice  as  many  dollars  as  there  are  boys,  and  together  they 
receive   138  dollars  :    how  many  women  were  there,  and 
how  many  boys  ? 

(36  women 
f  48  boys. 


EQUATIONS  OF   THE   SECOND   DEGREE.  173 

Of  Complete  Equations. 

122.  We  have  already  seen  (Art.  117),  that  a  complete 
equation  of  the  second  degree,  contains  the  square  of  the 
unknown  quantity,  the  first  power  of  the  unknown  quantity, 
and  known  terms. 

1.  If  we  have  the  complete  equation 
we  have,  by  transposing  and  reducing, 

and  by  dividing  by  3, 

ar2— 3ar=8, 
an  equation  containing  but  three  terms. 

2.  If  we  have  the  equation 

azx2 + 3abx + x2 = ex -f d, 
by    ollecting  the  coefficients  of  x2  and  x,  we  have 

(a2+l)x2+(3ab—c)x=d; 
and  dividing  by  the  coefficient  of  x2,  we  have 
3ab— c  d 


QUEST. — 122.  How  many  terms  does  a  complete  equation  of  the 
second  degree  contain1?  Of  what  is  the  first  term  composed  1  The 
second  1  The  third  ? 

16 


174  ELEMENTARY   ALGEBRA. 

If  we  represent  the  coefficient  of  x  by  2/>,  and  the  known 
term  by  </,  we  have 


an  equation  containing  but  three  terms. 

Hence,  we  see  that  every  complete  equation  of  the  second  de- 
gree can  be  reduced  to  an  equation  containing  but  three  terms 

123.  We  wish  now  to  show  that  there  are  four  forms 
under  which  this  equation  will  be  expressed,  each  depend- 
ing on  the  signs  of  2p  and  q. 

1  st.  Let  us  for  the  sake  of  illustration,  make 

2/>=-f-4,     and     ?—  +5: 
we  shall  then  have         #24-4a?:r_-5. 
2nd.  Let  us  now  suppose 

2p=—  4,     and     ^  =  +5: 
we  shall  then  have         x2  —  4x=5. 
3rd.  If  we  make 

2p=  +  4,     and     q^=—  5, 
we  have  x2+4x=  —  5. 

4th.  If  we  make 

2p=—  4,     and     q=  —  5, 
we  have  x2  —  4:X=  —  5. 


QUEST. — 123.  Under  how  many  forms  may  every  equation  of  the 
second  degree  be  expressed  1  On  what  will  these  forms  depend  ?  What 
are  the  signs  of  the  coefficient  of  x  and  the  known  term,  in  the  first 
form  1  What  in  the  second  ?  What  in  the  third  1  What  in  the  fourth  ' 
Repea*  the  four  forms. 


EQUATIONS  OF  THE  SECOND  DEGREE.      175 

We  therefore  conclude  that  every  complete  equation  of 
the  second  degree  may  be  reduced  to  one  of  these  forms  : 

#?  -f-  2pw  =  +  #,  1st  form. 

a:2  — 2px=+q,  2nd  form. 

xz-{-2px= — q,  3rd  form. 

x2—2px=—q,  4th  form. 

124.  REMARK. — If,  in  reducing  an  equation  to  either  of 
these  forms,  the  second  power  of  the  unknown  quantity 
should  have  a  negative  sign,  it  must  be  rendered  positive 
by  changing  the  sign  of  every  term  of  the  equation. 

125.  We  are  next  to  show  the  manner  in  which  the 
value  of  the  unknown  quantity  may  be  found.     We  have 
seen  (Art.  38),  that 


and  comparing  this  square  with  the  first  and  third  forms,  we 
see  that  the  first  member  in  each  contains  two  terms  of  the 
square  of  a  binomial,  viz  :  the  square  of  the  first  term  plus 


1  76  ELEMENTARY  ALGEBRA. 

called   completing  the  square.      Then,  by  extracting  the 
square  root  of  both  members  of  the  equation,  we  have 


and  x-\-p=  ±  -Y/— 

which  gives,  by  transposing  p, 

x=—p±  < 


126.  If  we  compare  the  second  and  fourth  forms  with 
the  square 


we  also  see  that  half  the  coefficient  of  x  being  squared  and 
added  to  both  members,  will  make  the  first  members  perfect 
squares.  Having  made  the  additions,  we  have 


Then,  by  extracting  the  square  root  of  both  members    we 
have 


and  x—  p=  ±  V—  q+p*  ; 

and  by  transposing  —  p,  we  find 


and  x=p  ±  ^/  — 


QUEST. — 120.  In  the  second  form,  how  do  you  make  the  first  mem- 
ber a  perfect  square  1 


EQUATIONS    OF    THE    SECOND    DEGREE.  177 

1  27.  Hence,  for  the  resolution  of  every  equation  of  the 
econd  degree,  we  have  the  foll9\ving 

RULE. 

I.  Reduce  the  equation  to  one  of  the  known  forma. 

II.  Take  half  the  coefficient  of  the  second  term,  square  it, 
and  add  the  result  to  both  members  of  the  equation. 

III.  Then  extract  the  square  root  of  both  members  of  the 
equation  ;  after  which,  transpose  the  known  term  to  the  second 
member. 

REMARK.  —  The  square  root  of  the  first  member  is  always 
equal  to  the  square  root  of  the  first  term,  plus  or  minus  half 
the  coefficient  of  x. 

EXAMPLES  IN  THE  FIRST  FORM. 

1.  What  are  the  values  of  x  in  the  equation 


If  we  first  divide  by  the  coefficient  2,  we  obtain 

x*+4x=32. 
Then,  completing  the  square, 

tf2 
Extracting  the  root, 


=  +  6  or  —6. 
Hence,  x=—  2  +  6=+4; 

or,  a?=  —  2  —  6  =  —  8. 


QUEST. — 127.  Give  the  general  rule  for  resolving  an  equation  or  the 
second  vlegree.     What  is  the  first  step  ?     What  the  second  1     What  the 
third 7      What  is  the  square  root  of  the  first  member  always  equal  to  ' 
16* 


1  73  ELEMENTARY  ALGEBRA. 

That  is,  in  this  form  the  smaller  root  is  positive,  and  the 
larger  negative. 

Verification. 

If  we  take  the  positive  value,  viz:    #=+4, 
he  equation  a?2 +4#=32 

gives  42+4x4  =  32: 

and  if  we  take  the  negative  value  of  x,  viz :    #=  —  8, 
the  equation  x2-\-4x=32 

gives  (—8)2+4(  — 8)^64-32-32. 

From  which  we  see  that  either  of  the  values  of   x,  viz : 
a:  =+4    or    0?= — 8,    will  satisfy  the  equation. 

2.  What  are  the  values  of  x  in  the  equation 
By  transposing  the  terms,  we  have 


and  by  reducing, 

4z2+24a:=108; 

and  dividing  by  the  coefficient  of  a;2, 


Now,  by  completing  the  square, 

z2-f-6.r  +9-36; 
extracting  the  square  root, 

v-{-?  =  ±'\/36  =  +  6  or  —  6: 
hence,  x=  +  6  —  3=1  +  3; 

or,  x=—  6—  3  =  —  9. 


EQUATIONS  OF  THE  SECOND  DEGREE.      179 

Verification. 
If  we  take  the  plus  root,  the  equation 

xz+6x=27 

gives  (3)*-r-6(3)=27; 

and  for  the  negative  root, 


gives  (  —  9)2+6(~9)  =  81  —  54=27. 

4.  What  are  the  values  of  x  in  the  equation 

a?2—  KXr-f  15=^  --  34z+155. 
o 

By  clearing  the  fractions,  we  hare 


by  transposing  and  reducing,  we  obtain 
4x*  f  120ar  =  700; 
then,  dividing  by  the  coefficient  of  a?2,  we  have 


and  by  completing  the  square, 

#2+30o:+225: 
and  by  extracting  the  square  root, 

x+15  =  ±  -v/400^4-20  or  —20 
Hence,  x=-\-5    or    —35. 

Verification. 
For  the  plus  value  of   a?,  the  equation 


gives  (5)24  30  x  5=25+  150  =  175 


1  80  ELEMENTARY  ALGEBRA. 

A.nd  for  the  negative  value  of  #,  we  have 

(_35)2_j_30(_35)  =  1225  —  1050  =  175. 

5.  What  are  the  values  of  a?  in  the  equation 


Clearing  the  fractions,  we  have 

10^—6^+9=96—  &r—  12a2+273  ; 
transposing  and  reducing, 

22x2+2a?=360; 
dividing  both  members  by  22, 

2         360 


2 
Add    {  —  )     to  both  members,  and  the  equation  becomes 


2      ,/iy      360      /I 
22^(22)  =-22-+ 

whence,  by  extracting  the  square  root, 


1  7360      /  1  \2 

^22= 

therefore, 


_     J_  .       /36<)    .  /  1  \2 


A  i          360 

and  ^==__ 


EQUATIONS  OF  THE  SECOND  DEGREE.      181 

It  remains  to  perform  the  numerical  operations.     In  the 

urst  place, \-( — \    must  be  reduced  to  a  single  num- 

22       \22/ 

her,  having  (22  )2  for  its  denominator. 

360     /  1  \2     360x22-f-l_7921 
'W>  ~22~  +  V22/  =         (22)2        ""(22)2 J 

extracting  the  square  root  of  792 1 ,  we  find  it  to  be  89 ; 
therefore, 


/360     /J_\2_     8£ 
V~22~      \22/  "^22* 


Consequently,  the  plus  value  of  x  is 

*=~: 

and  the  negative  value  is 


__J_     89_88 
~~22     22~22~   ' 


__1  __  £9__  45 
~~~~" 


that  is,  one  of  the  two  values  of  x  which  will  satisfy  the 
proposed  equation  is  a  positive  whole  number,  and  the  other 
a  negative  fraction. 

6.  What  are  the  values  of  x  in  the  equation 


7.  What  are  the  values  of  x  in  the  equation 


=5-?-— —  +  197. 
4        8 


Ans 


•  i:«8_, 


182  ELEMENTARY  ALGEBRA. 

8.  What  are  the  values  of  x  in  the  equation 


Ans. 
9.  What  are  the  values  of  x  in  the-  equation 


10.  What  are  the  values  of  a;  in  the  equation 


a?      jr__a?2_jr      13 
~~          ~  +      ' 


EXAMPLES    IN    THE    SECOND    FORM. 

1.  What  are  the  values  of  x  in  the  equation 


An*. 

x=-2$ 


By  transposing, 

x2  —  8x=19  — 10  =  9, 
then  by  completing  the  square 

x2— 8x+16  =  9-j-16=25, 
and  by  extracting  the  root 

x— 4=  rfc-v/25  =  +5     or     —5. 
Hence, 

a?=4-|-5  =  9     or     o?=4  —  5  =  —  1. 

That  is,  in  this  form,  the  largest  root   is  positive  and  the 
smaller  negative. 


EQUATIONS  OF  TUB  SECOND  DEGREE.      183 

Verification. 

If  we  take  the  positive  value  of  a;,  the  equation 

*>  -8a;=9     gives     (9)2—8x9  =  81  —  72  =  9; 
and  if  we  take  the  negative  value,  the  equation 

tf2--8a:  =  9     gives     (-1)2_8(-1)-1  +  8  =  9  ; 

from  which  we  see  that  both  values  alike  satisfy  the  equa- 
tion. 


2.  What  are  the  values  of  x  in  the  equation 

x2      x  x2 

_  +  __15=_  +  *-14i. 

By  clearing  the  fractions,  we  have 


—  177 

and  by  transposing  and  reducing 


and  dividing  by  the  co-efficient  of  a;2,  we  obtain 
T2      8r_i 

~Y 

Then,  by  completing  the  square,  we  have 


md  by  extracting  the  square  root, 


Hence, 


25          5  5 

=+      or  -- 


4,5  451 

+=+3    or    *=~=- 


184  ELEMENTARY  ALGEBRA. 

Verification. 
For  the  positive  value  of  a?,  the  equation 

--!•=' 

o 

gives  32  —  —  x  3=9—  8  =  1: 

3 

and  for  the  negative  value,  the  equation 


8  1       1,8 

TX  -T=T+T=1. 

3.  What  are  the  values  of  x  in  the  equation 


Clearing  the  fractions,  and  dividing  by  the  coefficient  of 
a:2,  we  have 


Completing  the  square,  we  have 


then,  by  extracting  the  square  root,  we  have 


hence, 


7  7 

-=±=+      or  -"; 


1    ,   7       9      lt  17  5 

T4T=T=li,    or    ,=---= -Y. 


EQUATIONS  OF  THE  SECOND  DEGREE.      185 

Verification 
If  we  take  the  positive  value  of  #,  the  equation 


gives  (!})«—    xl}=2}-l=lj: 

and  for  the  negative  value,  the  equation 


*      2  5      25      10     45 


4.  What  are  the  values  of  x  in  the  equation 
=l8ab  —  l8bz  T 


By  transposing,  changing  the  signs,  and  dividing  by  2,  it 
becomes 

x2—  ax=2a2— 

whence,  completing  the  square, 


4          4 
extracting  the  square  root, 


*=T 


0 

Now,  the  square  root  of     — 9ab+9b2,  is  evidently 


~— 36.     Therefore, 


?=     2a— 36, 

'  =  —  ±1—  -3*),     <*       J  ,    or 

f   #:=:  —     a-f-30. 

17 


186  ELEMENTARY  ALGEBRA. 

What  will  be  the  numerical  values  of  x,  if  we  suppose 
a  =  6  and  b=l  ? 

5.  What  are  the  values  of  x  in  the  equation 

.Lt_4_38+2a-- 1*2  =  45- 3*2+4*  ? 

o  O 

C  x=     7,12  >  to  within 
.Aws.     <  > 

lx=—  5,73*       0,01. 

6.  What  are  the  values  of  x  in  the  equation 


7.  What  are  the  values  of  x  in  the  equation 

T 

Ans.     \x  =     S' 

(  ar=  —  4. 

8.  What  are  the  values  of  x  in  the  equation 

xz 
2 

f  x Q 

Ans.     < 

9.  What  are  the  values  of  x  in  the  equation 

(  a* —     2a-L-b. 
An*     ] 

(  x=—b. 

10.  What  are  the  values  of  x  in  the  equation 


~n2— mz 
Ans. 


EQUATIONS    OF    THE    SECOND    DEGREE.  187 

EXAMPLES  IN  THE  THIRD  FORM. 

1.   What  are  the  values  of  x  in  the  equation 


First,  by  completing  the  square,  we  have 
x*+4x+4  =  —  3  +  4=1  ; 
and  by  extracting  the  square  root, 

x+2=±  -v/T=  +  l    or    —  1  : 
nence,     x=—  2-j-l  =  —  1  ;    or    x=—  2  —  1  =—  3 
That  is,  in  this  form  both  the  roots  are  negative. 

Verification. 
If  we  take  the  first  negative  value,  the  equation 

X*  +  4X=  —  3 

gives  (-l)»+4(-l)  =  l-4  =  -3; 

and  by  taking  the  second  value,  the  equation 

x*+4x=—  3 

gives  (_3)2_|_4(_3)-9_i2  =  —  3  : 

hence,  both  values  of  x  satisfy  the  given  equation. 
2    What  are  the  values  of  *  in  the  equation 


By  transposing  and  reducing,  we  have 
-*2-ll;r=28; 

then  since  the  coefficient  of  the  second  power  of  x  is  nega- 
tive, we  change  the  signs  of  all  the  terms  which  gives 

r=—  28, 


1 88  ELEMENTARY  ALGEBRA. 

then  by  completing  the  square 

a?2+lla?+30,25=2,25, 
hence, 

ar-f  5,5=  ±-y/2,25  =  -|-l,5     or     — 1,5. 
onsequently, 

x= — 4     or     x= — 7. 

3.  What  are  the  values  of  x  in  the  equation 

<  x=—  2 
Ans.   1 

I  x=z — 5. 

4.  What  are  the  values  of  x  in  the  equation 

2#2+8ar=  — 2j x. 

3 

Ans.  { 

U=-J. 

5.  What  are  the  values  of  x  in  the  equation 

5 

6.  What  are  the  values  of  x  in  the  equation 

2_4_JL    —— 

~T*  ~"  2 

7.  What  are  the  values  of  x  in  the  equation 

— a:2-*-  7a?-{-20:= a?2 — Ha? — 60. 

9  9 

Ans.    <  ~~ 

?=  -   8. 


EQUATIONS    OF    THE    SECOND    DEGREE.  189 

8.  What  are  the  values  of  a?  in  the  equation 


,  *•=  —  8 
Ans. 


9.  What  are  the  values  of  x  in  the  equation 


Ans.  I  *       " 

I  a:=— T%. 

lO.  What  are  the  values  of  a;  in  the  equation 


1  1  .  What  are  the  values  of  x  in  the  equation 
:—  90=—  93. 


=—  3 
Ans. 


EXAMPLES    IN    THE    FOURTH    FORM. 

1  .  What  are  the  values  of  x  in  the  equation 

x2  —  8x=  —  7. 
By  completing  the  square  we  have 

x*—8x+l6=i—  7+16=9; 
then  by  extracting  the  square  root 

x—  4  =  ±-y/9^:  +  3     or     —3; 


hence, 

»=+7     or     a:  =+1. 

That  is,  in  this  form,  both  the  roots  are  positive. 
17* 


190  ELEMENTARY  ALGEBRA. 

Verification. 
If  \i  e  take  the  largest  root,  the  equation 

x*—8x=—7     gives     72  —  8  x  7  =  49-56=  -7; 
and  for  the  smaller,  the  equation 

a2  —  8a?=—  7     gives     I2  —  8  x  1  =  1—  8  =  —  7  : 
hence,  both  of  the  roots  will  satisfy  the  equation. 
2.  What  are  the  values  of  x  in  the  equation 


—  I-'-*2  +  3#—  10  =  IJz2— 

By  clearing  the  fractions,  we  have 

.  ~   —3X2+GX—  20=3*2—  36a?+40; 
then  by  collecting  the  like  terms 

—  6#2+42a:=60  ; 

then  by  dividing  by  the  coefficient  of  a:2,  and  at  the  same 
time  changing  the  signs  of  all  the  terms,  we  have 

x2—  7x=  —  10. 
By  completing  the  square,  we  have 

xz—lx+  12,25  =2,25, 
and  by  extracting  the  square  root  of  both  members, 

a?—  3,5=db  V2,25  =  +  l,5     or     —1,5. 
hence, 

*=3,5-{-  1,5  =  5,     or     x=3,5  —  1,5=2. 


EQUATIONS    OF    THE    SECOND    DEGREE.  191 

Verification. 

If  we  take  the  larger  root,  the  equation 

x*—7x=  — 10     gives     52— 7x5=25— 35  =  - 10  ; 
.  and  if  we  take  the  smaller  root,  the  equation 

X*  —  7x=  — 10     gives     22  —  7x2=4  — 14.=  — 10. 
3.  What  are  the  values  of  x  in  the  equation 

—  3x+2xz+l  =  l7£x— 2x2  —  3,     \ 
By  transposing  and  collecting  the  terms,  we  have 

4a;2— 20|*=  —  4; 
then  dividing  by  the  coefficient  of  x2  we  have 

*2-5j*=-l. 

By  completing  the  square,  we  obtain 

«HS-l.+5-Sf. 

and  by  extracting  the  root 

/T44~        12  12 

*2_2i  =  ±x/_  =  +_     or     -_; 

hence, 

*=21+T=5'    or'    •=8*-T=f- 

Verification. 

If  we  take  the  larger  root,  the  equation 

a;2  — 5|-a?=— 1     gives     52  — 5|  x5=25— 26  =  — 1 
suid  if  we  take  the  smaller  root,  the  equation 


1  92  ELEMENTARY  ALGEBRA. 

4.  What  are  the  values  of  x  in  the  equation 

Ans. 

5.  What  are  the  values  of  x  in  the  equation 

1 
7 

Ans. 

6.  What  are  the  values  of  x  in  the  equation 

81  11, 


7.  What  are  the  values  of  x  in  the  equation 


Ans. 


8.  What  are  the  values  of  x  in  the  equation 

17  xz  2r2 

--+100=—- 
5  5 


An,. 

x= 


9.  What  are  the  values  of  x  in  the  equation 

7a* 

~~    3 


Ans- 


10.  What  are  the  values  of  x  in  the  equation 

JLi 

10 


EQUATIONS  OF  THE  SECOND  DEGREE.      193 


Properties  of  the  Roots. 

128.  We  have  thus  far,  only  explained  the  methods  of 
finding  the  roots  of  an  equation  of  the  second  degree.     We 

.  are  now  going  to  show  some  of  the  properties  of  these  roots 

The  first  form. 

129.  The  first  form 


gives  1st   root  x=  —  p  f 

2nd  root  x=  —  p  — 

and  their  sum  =  —  2p. 

Since,  in  this  form  q  is  supposed  positive,  the  quantity 
q+p2  under  the  radical  sign  will  be  greater  than  j>2,  and 
hence  its  root  will  be  greater  than  p.  Consequently  the 
first  root,  which  is  equal  to  the  difference  between  p  and 
the  radical,  will  be  positive  and  less  than  \/q-{-p2.  In  the  second 
root,  p  and  the  radical  have  the  same  sign  ;  hence,  the 
second  root  will  be  equal  to  their  sum  and  negative.  If  we 
multiply  the  two  roots  together,  we  have 

—  p  -f- 
-p  - 


+P2-pV<l+P2 


Product  equal  to      .....      —  q. 


QUKST. — 129.  In  the  first  form,  have  the  roots  the  same  or  contrary 
signs'!  What  is  the  sign  of  the  first  root]  What  of  the  second  I 
Which  is  the  greater  1  What  is  ».heir  sum  equal  to  7  What  is  their 
product  equal  to  1 


194  ELEMENTARY  ALGEBRA. 

Hence  we  conclude, 

1st.   That  in  the  first  form  one  of  the  roots  is  always  posi 
ttve,  and  the  other  negative. 

2nd.  That  the  positive  root  is  numerically  less  than  tht 
negative. 

3rd.   That  the  sum  of  the  two  roots  is  equal  to  the  coefficient 
of  x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.   That  the  product  of  the  two  roots  is  equal  to  the  known 
term  in  the  second  member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1.  In  the  equation 


we  find  the  roots  to  be  4  and  —5.     Their  sum  is   —1, 
and  their  product  —20. 

2.  In  the  equation 


we  find  the  roots  to  be  1  and  —3.     Their  sum  is  equal  to 
—2,  and  their  product  to   —3. 

3.  The  roots  of  the  equation 

x*+x  —  90, 

are  _L_9  and  —  10.     Their  sum  is   —  1,  and  their  product 
—90. 

4.  The  roots  of  the  equation 


are  6  and  —10.     Their  sum  is    -4,  and  their  product  is 
-60. 


EQUATIONS    OF    THE    SECOND    DEGREE.  195 

!  -et.  these  principles  be  applied  to  each  of  the  examples 

Udder    "  EXAMPLES    IN    THE    FIRST    FORM." 

Second  Form. 
13O.  The  second  form  is, 


and  by  resolving  the  equation  we  find 
1st  root,  ix=  -\-p-\- 

2nd  root, 
and  their  sum 

In  this  form,  the  first  root  is  positive  and  the  second 
negative.     Tf  we  multiply  the  two  roots  together,  we  have 


Hence  we  conclude, 

1st.   That  in  the  second  form  one  of  the  roots  is  positive 
and  the  other  negative. 

2nd.    That  the  positive  root  is  numerically  greater  than  th& 
negative. 

3rd.   That  the  sum  of  the  roots  is  equal  to  the  coefficient  of 
X  in  the  second  term,  taken  with  a  contrary  sign. 

4th.   That  the  product  of  the  roots  is  equal  to  the  known 
t?rm  in  the  second  member,  taken  with  a  contrary  sign. 


QUEST. — 130.  What  is  the  sign  of  the  first  root  in  the  second  form  1 
\\Trat  is  the  sign  of  the  second  1  Which  is  the  greater  1  What  is  their 
sum  equal  to  1  What  is  their  product  equal  to  1 


196  ELEMENTARY  ALGEBRA. 

EXAMPLES. 

1.  The  roots  of  the  equation 

xz—x=I2, 

are    +4  and  —  3.     Their  sum  is    -f-l>  and  their  product 
—  12. 

2.  The  roots  of  the  equation 


are  +10  and  —  —  .     Their  sum  is  9j^,  and  their  product 
is  —1. 

3.  The  'roots  of  the  equation 


are  +8  and  —2.     Their  sum  is   +6,  and  their  product 
is   —16. 

4.  The  roots  of  the  equation 


are   -{-16  and  —  5.     Their  sum  is  -|-  1  1  ,  and  their  product 
is   —80. 

Let  these  principles  be  applied  to  each  of  the  examples 
under  "  EXAMPLES  IN  THE  SECOND  FORM." 

Third  Form. 
131.  The  third  form  is, 


and  by  resolving  the  equation  we  find, 
1st  root,  #:= 


2nd  root,  x=—  p—  i/ 

Their  sum  is  =—  2p 


EQUATIONS    OF    THE    SECOND    DEGREE.  197 

In  tins  form,  the  quantity  under  the  radical  being  less 
than  p2j  its  root  will  be  less  than-  p  :  hence  both  the  roots 
will  be  negative,  and  the  first  will  be  numerically  the  least. 

If  we  multiply  the  roots  together,  we  have 


Hence  we  conclude, 

1  st.   That  in  the  third  form  both  the  roots  are  negative. 
2nd.   That  the  first  root  is  numerically  less  than  the  second. 

3rd.   That  the  sum  of  the  two  roots  is  equal  to  the  coefficient 
of  x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.   That  the  product  of  the  roots  is  equal  to  the  known 
term  in  the  second  member  ',  taken  with  a  contrary  sign. 

EXAMPLES. 
1.  The  roots  of  the  equation 

#=:—  20, 


are   —4  and  —5.     Their  sum  is   —9,  and  their  product 

+  20. 

2.  The  roots  of  the  equation 

#2+13*:=—  42, 


are    —6  and  —7.     Their  sum  is  —13,  and  their  product 

+42. 


QUEST. — 131.  In  the  third  form,  what  are  the  signs  of  the  roots  ? 
Which  root  is  the  least  1  What  is  the  sum  of  the  roots  equal  to  1 
What  is  their  product  equal  to  1 

18 


1  98  ELEMENTARY  ALGEBRA. 

3.  The  roots  of  the  equation 


are   —  —  and   —2.     Their  sum  is   —2|,  and  their  product 


4.  The  roots  of  the  equation 

=—  6, 


are   —2  and  —  3.      Their  sum  is   —5,  and  their  product 
is  +6 

Let  these  principles  be  applied  to  each  of  the  examples 
under  "  EXAMPLES  IN  THE  THIRD  FORM." 

Fourth  Form. 

132.  The  fourth  form  is, 

x2—  2px=  —  q  ; 

and  by  resolving  the  equation  we  find, 
1st  root,  x=p-{--\/  — 

2nd  root, 

Their  sum  is 

In  this  form,  as  well  as  in  the  third,  the  quantity  under 
the  radical  being  less  than  p2,  its  root  will  be  less  than  p  : 
hence  both  the  roots  will  be  positive,  and  the  first  will  be 
the  greatest. 

If  we  multiply  the  two  roots  together,  we  have 

—  -1-7- 


t 
EQUATIONS    OF    THE    SECOND    DEGREE.  199 

Hence  we  conclude, 

1  st.    That  in  the  fourth  form  both  the  roots  are  positive. 

2nd.   That  the  first  root  is  greater  than  the  second. 

3rd.  That  the  sum  of  the  roots  is  equal  to  the  coefficient  of 
x  in  the  second  term,  taken  with  a  contrary  sign. 

4th.  That  the  product  of  the  roots  is  equal  to  the  known 
term  in  the  second  member,  taken  with  a  contrary  sign. 

EXAMPLES. 

1  .  The  roots  of  the  equation 

*2  —  7x=  —  12, 

are     +4    and    +3.     Their   sum  is    +7    and   their   pro- 
duct   +  12. 

2.  The  roots  of  the  equation 

*2—  14*=  -24, 

are    +12    and    +2.     Their  sum  is    +14   and  their  pro- 
duct   +  24. 

3.  The  roots  of  the  equation 


are    +18    and   +2.     Their  sum  is    +20   and  their  pro- 
duct   +  36. 

4.  The  roots  of  the  equation 

x2—  \7x=—42, 

are    +14    and    +3.     Their  sum  is    +17    and   thoir  pro 
duct    +42. 


QUEST. — 132.  In  the  fourth  form,  what  are  the  signs  of  the  roots  1 
Which  root  is  the  greatest  1  What  is  the  sum  of  the  roots  equal  to  ? 
What  is  their  product  e<]ual  to  7 


200  ELEMENTARY  ALGEBRA. 

133.  In  the  third  and  fourth  forms  the  values  of  x  some- 
times become  imaginary,  and  in  such  cases  it  is  necessary 
to  know  how  the  results  are  to  be  interpreted. 

If  we  have  q>p2,  that  is,  if  the  known  term  is  greater 
than  half  the  coefficient  ofx  squared,  it  is  plain  that  ^/ —q+p2 
will  be  imaginary,  since  the  quantity  under  the  radical 
will  be  negative.  Under  this  supposition  the  values  of  x 
in  the  third  and  fourth  forms  will  be  imaginary. 

We  will  now  show  that,  when  in  the  third  and  fourth 
forms,  we  have  q^>j)2,  the  conditions  of  the  question  will  be 
incompatible  with  each  other. 

134.  Before  showing  this  it  will  be  necessary  to  estab- 
lish a  proposition  on  which  it  depends  :  viz. 

If  a  given  number  be  decomposed  into  two  parts  and  those 
parts  multiplied  together,  the  product  will  be  the  greatest  pos- 
sible when  the  parts  are  equal. 

Let  2p  be  ihe  number  to  be  decomposed,  and  d  the  differ- 
ence of  the  parts.  Then 

p  +-o- =         the  greater  part  (page  80,  Ex.  7.) 

and         p =         the  less  part. 

2 

d2 
and        p2 =  P,     their  product  (Art.  4O.) 

Now  it  is  plain  that  P  will  increase  as  d  diminishes,  and 
that  it  will  be  the  greatest  possible  when  d=0  :  that  is, 

p  xp  =p2     is  the  greatest  product. 


QUEST. — 133.  In  which  fr  .as  do  the  values  of  x  become  imaginary7 
When  will  the  values  of  x  be  imaginary  ?  Why  will  the  values  of  x  be 
then  imaginary  1 


EQUATIONS  OF  THE  SECOND  DEGREE.      201 

Now,  since  in  the  equation 


2p  is  the  sum  of  the  roots,  and  q  their  product,  it  follows 
that  q  can  never  be  greater  than  p2.  The  conditions  of  the 
equation,  therefore,  fix  a  limit  to  the  value  of  q,  and  if  we 
make  q^p2,  we  express  by  the  equation  a  condition  which 
cannot  be  fulfilled,  and,  this  contradiction  is  made  apparent 
by  the  values  of  x  becoming  imaginary.  Hence  we  may 
conclude  that, 

When  the  values  of  the  unknown  quantity  are  imaginary, 
the  conditions  of  the  question  are  incompatible  with  each  other. 

EXAMPLES. 

1.  Find  two  numbers  whose  sum.  shall  be  12  and  pro- 
duct 46. 

Let  x  and  y  be  the  numbers. 

By  the  1st  condition,  x-\-  y=l2  ; 

and  by  the  2d,  xy  =  46. 

The  first  equation  gives 

x  =  l2—y. 
Substituting  this  value  for  x  in  the  second,  we  have 

12y-y2  =  46; 
and  changing  the  signs  of  the  terms,  we  have 

*—l2=  —  46. 


QUKST. — 134.  What  is  the  proposition  demonstrated  in  Article  134] 
If  the  conditions  of  the  question  are  incompatible,  how  will  the  values 
of  the  unknown  quantit)  be 1 

18* 


202  ELEMENTARY  ALGEBRA. 

Then  by  completing  the  square 

yz—l2y+36=—  46  +  36  =  —  10 
which  gives  y  =  6  +  -\/  —  10, 

and  y  —  6—  -y/  —  10; 

both  of  which  values  are  imaginary,  as  indeed  they  should 
be,  since  the  conditions  are  incompatible. 

2.  The  sum  of  two  numbers  is  8,  and  their  product  20  • 
what  are  the  numbers  ? 

Denote  the  numbers  by  x  and  y. 
By  the  first  condition, 


and  by  the  second,  a?y=20. 

The  first  equation  gives 

cc  =  8—  y 
Substituting  this  value  of  x  in  the  second,  we  have 

8y—  y2=20  ; 
changing  the  signs,  and  completing  the  square,  we  have 


and  by  extracting  the  root, 

y=4-{--v/^4    and    y=4  — -y/— 4. 
These  values  of  y  maybe  put  under  the  forms  (Art.  1O6) 

3.  What  are  the  values  of  x  in  the  equation 
—  -10. 

Ans. 


EQUATIONS  OF  THE  SECOND  DEGREE.      203 

Examples  with  more  than  one  unknown  quantity. 

tofind  * and  y' 

By  transposing  y   in  the  first  equation,  we  have 

a?=14— y  ; 
and  by  squaring  both  members, 

*2=196  — 28y+y2. 
Substituting  this  value  for  a?2  in  the  2nd  equation,  we  have 

196— 28y+y2+y2=100  ; 
from  which  we  have 

y2— 14y=  —  48  ; 
and  by  completing  the  square, 

y2  — 14y+49  =  l  ; 
and  by  extracting  the  square  root, 

y— 7=±yT=+l  or  —1: 
hence,  y— 7+1=8,    or    y=7  — 1=6. 

If  we  take  the  larger  value,  we  find  a; =6  ;  and  if  we 

take  the  smaller,  we  find  a? =8. 

Verification. 

For  the  largest  value,  y  =  8,  the  equation 
x+y  =  14     gives     6  +  8  =  14; 
and  #24-y2  =  100     gives     36+64  =  100. 

For  the  value  y  =  6,  the  equation 

#+yr=14     gives     8  +  6  =  14; 
and  a;2+y2  =  100     gives     64  +  36  =  100. 

Hence,  both  sets  of  values  will  satisfy  the  given  equation 


204  ELEMENTAEY  ALGEBRA. 

/     y>     —  —  */     —       3      I 

2.   Given     <  \     to  find  x  and  y. 

(  x2—  y2  =  45  ) 

Transposing  y  in  the  first  equation,  we  have 


and  then  squaring  both  members, 


Substituting  this  value  for  x2  in  the  second  equation,  we 
have 


whence  we  have 

6y=36    and    y=6. 
Substituting  this  value  of  y  in  the  first  equation,  we  have 

#-6  =  3, 
and  consequently  x  =3-}-  6  =9. 

Verification. 

x—y  =  3     gives     9  —  6  =  3; 
and  a:2—  y2  =  45     gives     81—36=45. 

3-    Given      \  Jt5+2y*  =  40  i      *°  find  X  a"d  '• 
Subtracting  the  first  equation  from  the  second,  we  have 

2y2=18, 

which  gives  y2=9» 

and  y=-f-3    01    —3. 

Substituting  the  plus  value  in  the  first  equation,  we  have 
2 


EQUATIONS  OF  THE  SECOND  DEGREE.      205 

from  which  we  find 

x=+2     and     #=  —  11. 

If  we  take  the  negative  value,  y=  —  3,  we  have  from  the 
first  equation, 

a;2—  9#=22  ; 
from  which  we  find 

and     a=—  2. 


For  the  values  y=+3  and  x  =+2,  the  equation 


gives  22+3x2x3=4  +  18=22: 

and  for  the  second  value,  ar=  —  11,  the  same  equation 


gives         (—  ll)2+3x  -11x3  =  121—99=22. 

If  now  .we  take  the  second  value  of  y,  that  is,  y=  —  3 
and  the  corresponding  values  of  ar,  viz,  #=-{-11,  and 
x=  —  2  ;  for  #=  -f-  1  1  ,  the  equation 


gives  H2+3  x  11  x  -3  =  121  -99=22  ; 

and  for  x=  —  2,  the  same  equation 


gives  (-2)2+3  X  —2  X  —3=4+18=22. 

4.  Given  <  x  +y  ~\-z  =  7     (2)  >  to  find  xt  y,  and  z. 
(  x*+vziz2=2i      (3)  j 


206  ELEMENTARY  ALGEBRA. 

Transposing   y   in  the  second  equation,  we  have 

*+*=7-y    (4); 
then  squaring  the  members,  we  have 

x*+2xz+z2=49  —  14y+y2. 

If  now  we  substitute  for   2xz   its  value  taken  from  the 
first  equation,  we  have 

a:2+2y2+;s2=49  —  Hy+y2  ; 
and  cancelling  y2  in  each  member,  there  results 


But,  from  the  third  equation  we  see  that  each  member  of 
the  last  equation  is  equal  to  21  :  hence 

49  —  14y=21, 
and  14y=49—  21=28. 

28 
hence,  y=—  =2. 

Placing  this  value  for  y  in  equation  (1)  gives 

xz=4  j 
and  placing  it  in  equation  (4)  gives 

x+z=5,     and     x  =  5—z. 

Substituting  this  value  of  x  in  the  previous  equation,  we 
obtain 

5z—z*=4     or     z*—5z=—  4; 
and  by  completing  the  square,  we  have 
s2—  5^+6,25=2,5, 

and  z—2,5  =  ±«\/2£  =  +  1,5     or     —1,5; 

nence,       z=  2,5+1,5=4     or     *=  -1-2,5  —  1,5  =  1. 


EQUATIONS    OF    THE    SECOND    DEGREE  207 

If  we  take  the  value 

z=4,     we  find     x=l  : 
if  we  take  the  less  value 

*=1,     we  find     x=4. 

3.  Given     x  +  ^fxy-\-y  —   19 


—   19  ) 

\      to 
=l33  S 


o 
and        x*+     xy+yz 

Dividing  the  second  equation  by  the  first,  we  have 

«—  -/ay+  y=  7 

but  x+</xy+  y=l9 


nence,  by  addition,  2x  +2y=26 

or  a  4-  y=13 

and  substituting  in  1st  equa.     y^y-f-13  =  19 

or  yricy"=   6 

and  by  squaring  a:y=36 

From  2d  equation,  x2+a:y  +y2=133 

and  from  the  last  3xy        =108 


Subtracting                                     a;2—  2ary-f  y*=  25 

hence,                                                     ar  —  y=db  5 

but                                                             a?4-y=  13 

hence  o?=9   or   4  ;     and  y=4   or   9. 

6.  Given  the  sum  of  two  numbers  equal  to  a,  and  the 
sum  of  their  cubes  equal  to  c,  to  find  the  numbers 


By  the  conditions 


ix+y=a 


208 


ELEMENTARY  ALGEBRA. 


Putting        x=s-\-z,     and     y=s — z,     we  have 


a=2s,     or     s=  —  ; 
2 


and 


3szz-\-3sz2—z3. 


hence,  by  addition,    x3  +  y3 = 2s3  -f  6sz2 = c, 
whence  zz="     ~"      and     z=. 


or 


;     and     y= 


c—: 
6~* 

c — \ 
6^~' 


or  by  putting  for  s  its  value, 


and 


NOTE.  —  What  are  the  numbers  when   a=5    and   c=35. 
What  are  the  numbers  when   a=9   and   c=243. 


QUESTIONS. 

1.  Find  a  number  such,  that  twice  its  square,  added  to 
three  times  the  number,  shall  give  65. 

Let  x  denote  the  unknown  number.  Then  the  equation 
of  the  problem  will  be 


whence 


65      9 


3      23 


EQUAUON&    OF    THE    SECOND    DEGREE.  209 

Therefore, 

3    ,  23  3      23          13 

.=  -T+T=5,    and    *=-T-T=_T. 

Both  these  values  satisfy  the  question  in  its  algebraic 
sense.     For, 


/     13\2  13      169      39      130 

and      2(__)  +3  x  --=_-  y=_=65. 

REMARK.  —  If  we  wish  to  restrict  the  enunciation  to  its 
arithmetical  sense,  we  will  first  observe,  that  when  x  is 
replaced  by  —  x,  in  the  equation  2x2.-}-3x=65,  the  sign  of 
the  second  term  3x  only,  is  changed,  because  (  —  x)2=x*. 

q        oq 

Therefore,  instead  of  obtaining  x=  —  T^~T>  we  s^ou^ 

3      23  13 

find   x=  —  ±  —  ,  or  x=  —  ,  and  x=  —  5,  values  which  only 

differ  from  the  preceding  by  their  signs.     Hence,  we  may 

13 
say  that  the  negative  solution  --  ,    considered    indepen 

dently  of  its  sign,  satisfies  this  new  enunciation,  viz  :  To 
find  a  number  such,  that  twice  its  square,  diminished  by  three 
times  the  number,  shall  give  65.  In  fact,  we  have 


13      169      39 


REMARK.  —  The  root  which  results  from  giving  the  plus 
sign  to  the  radical,  generally  resolves  the  question  both 
in  its  arithmetical  and  algebraic  sense,  while  the  second 
root  resolves  it  in  its  algebraic  sense  only. 

19 


210  ELEMENTARY   A.LGEBRA. 

Thus,  in  the  example,  it  was  required  to  find  a  number, 
of  which  twice  the  square  added  to  three  times  the  number 
shall  give  65.  Now,  in  the  arithmetical  sense,  added  means 
increased  ;  but  in  the  algebraic  sense  it  implies  diminution, 
when  the  quantity  added  is  negative.  In  this  sense,  the 
second  root  satisfies  the  enunciation. 

2.  A  certain  person  purchased  a  number  of  yards  of  cloth 
for  240  cents.  If  he  had  received  3  yards  less  of  the  same 
cloth  for  the  same  sum,  it  would  have  cost  him  4  cents  more 
per  yard.  How  many  yards  did  he  purchase  ? 

Let         x=       the  number  of  yards  purchased. 

240 
Then     will  express  the  price  per  yard. 

If,  for  240  cents,  he  had  received  3  yards  less,  that  is 
x—3  yards,  the  price  per  yard,  under  this  hypothesis,  would 

240 
have  been  represented  by       • — .     But,  by  the  enunciation, 

OC O 

this  last  cost  would  exceed  the  first  by  4  cents.     Therefore, 
we  have  the  equation 

240       240  _ 

^=3        ~:  :4; 

whence,  by  reducing    a;2— 3a?=180, 


therefore  a?=15     and     sc  =  —  12. 

The  value  x  =15  satisfies  the  enunciation;  for,  15  yards 

240 
of  240  cents  gives     ,     or  16  cents  for  the  price  of 

I  O 

one  yard,  and  12  yards  for  240  cents,  gires  20  cents  for  the 
prirr  of  one  yard,  which  exceeds  16  by  4. 


EQUATIONS  OF  THE  SECOND  DEGREE.      21  1 

As  to  the  second  solution,  we  can  form  a  new  enuncia- 
tion, with  which  it  will  agree.  For,  going  back  to  the 
equation,  and  changing  x  into  — #,  it  becomes 

240         240  240       240 


-x-3      -x  x        x+3 

an  equation  which  may  be  considered  the  algebraic  transla- 
tion of  this  problem,  viz  :  A  certain  person  purchased  a  num- 
ber of  yards  of  cloth  for  240  cents :  if  he  had  paid  the  same 
sum  for  3  yards  more,  it  would  have  cost  him  4  cents  less  per 
yard.  How  many  yards  did  he  purchase  ? 

Ans.  07=12,  and  x=  — 15. 

3.  A  man  bought  a  horse,  which  he  sold  after  some  time 
for  24  dollars.  At  this  sale,  he  loses  as  much  per  cent, 
upon  the  price  of  his  purchase  as  the  horse  cost  him. 
What  did  he  pay  for  the  horse  ? 

Let  x  denote  the  number  of  dollars  that  he  paid  for  the 
horse,  #  —  24  will  express  the  loss  he  sustained.  But  as 

he  lost  x  per  cent,  by  the  sale,  he  must  have  lost    

i  CO 

upon  each  dollar,  and  upon  x  dollars  he  loses  a  sum  de- 
noted by  ;  we  have  then  the  equ  :on 

x2 

=o?—24,     whence     x2  —  10(L  =—2400. 


100 

and  ar=50±-v/2500-  >"V>_ 

Therefore,  a; =60    and    a  =40. 

Both  of  these  values  satisfy  the  question. 

For,  in  the  first  place,  suppose  the  man  gave  $60  for  the 
horse  and  sold  him  for  24,  he  loses  36.  Again,  from  the 
enunciation,  he  should  lose  60  per  cent,  of  60,  that  is, 


212  ELEMENTARY  ALGEBRA. 

of  60,  or   — TTTTT-J    which    reduces    to    36  ;    there- 


100  100 

fore,  60  satisfies  the  enunciation. 

Had  he  paid  $40,  he  would  have  lost  $16  by  the  sale  ; 

40 
for,  he  should  lose  40  per  cent,  of  40,  or    40  x  --  ,    which 

reduces  to  16  ;  therefore,  40  verifies  the  enunciation. 

4.  A  man  being  asked  his  age,  said  the  square  root  oi 
my  own  age  is  half  the  age  of  my  son,  and  the  sum  of 
our  ages  is  80  years  :  what  was  the  age  of  each  ? 

Let     x=     the  age  of  the  father. 

y=     that  of  the  son. 
Then  by  the  first  condition 


.. 

and  by  the  second  condition 

z+y  =  80. 
If  we  take  the  first  equation 

^f 

and  square  both  members,  we  have 

,=£ 

If  we  transpose   y   in  the  second,  we  have 

x=80—y: 
from  which  we  find 


y=— 

by  taking  the  plus  root,  which  answers  to  the  question  in 
its  arithmetical  sense.  Substituting  this  value,  we  find 
oc=64.  j  J  Father's  age  64 

nS'  (  Son's  16 


EQUATIONS    OF    THE    SECOND    DEGREE.  213 

5.  Find  two  numbers,  such  that  the  sum  of  their  pro- 
ducts by  the  respective  numbers  a  and  &,  may  be  equal  to 
2s,  and  that  their  product  may  be  equal  to  p. 

Let  x  and  y  be  the  required  numbers,  we  have  the  equa- 
tions 

ax -\- by  =  2s. 
and  xy=p. 

From  the  first  y=— — 7 » 

whence,  by  substituting  in  the  second,  and  reducing, 
aoc2—2sx= — bp. 

Therefore, 
and  consequently, 


_   S          1         -g ,-— 

b       b 

This  problem  is  susceptible  of  two  direct  solutions,  be- 
cause s  is  evidently  >  -\/s2—abp  ;  but  in  order  that  they 
may  be  real,  it  is  necessary  that  s2>  or  =abp. 

Let  a=b=l.  ;  the  values  of  x  and  y  reduce  to 

Whence  we  see,  that  the  two  values  of  x  are  equal  to 
those  of  y,_taken  in  an  inverse  order ;  which  shows,  that  if 
$4-  -\/s2—p  represents  the  value  of  x,  s —  -\/s2—p  will  re- 
present the  corresponding  value  of  y,  and  reciprocally. 

This  circumstance  is  accounted  for,  by  observing  that  in 
this  particular  case  the  equations  reduce  to 


19* 


214  ELEMENTARY  ALGEBRA. 

and  then  the  question  is  reduced  to,  finding  two  numbers  of 
which  the  sum  is  2s,  and  their  product  p,  or  in  other  Mrords 
to  divide  a  number  2s,  into  two  such  parts,  that  their  produc. 
may  be  equal  to  a  given  number  p. 

Let  us  now  suppose 


2s=l4     and    p  =  48: 
what  will  then  be  the  values  of  x  and  y  ? 


(  x= 
Ans.     < 

(     = 


x  =  8  or  6 
y=6  or  8 

6.  A  grazier  bought  as  many  sheep  as  cost  him  £60,  and 
after  reserving  fifteen  out  of  the  number,  he  sold  tho  re- 
mainder for  j£54,  and  gained  2s.  a  head  on  those  he  sold  : 
how  many  did  he  buy  ?  Ans.  75. 

7.  A  merchant  bought  cloth  for  which  he  paid  j£33   15.sv, 
which  he  sold  again  at  £2  8s.  per  piece,  and  gained  by  the 
bargain  as  much  as  one  piece  cost  him  :  how  many  pieces 
did  he  buy?  Ans.   15. 

8.  What  number  is  that,  which,  being  divided  by  the  pro- 
duct of  its  digits,  the  quotient  is  3  ;  and  if  18  be  added  to 
it,  the  digits  will  be  inverted  ?  Ans.  24, 

9.  To  find  a  number,  such  that  if  you  subtract  it  from  1 0, 
and  multiply  the  remainder  by  the  number  itself,  the  product 
shall  be  21.  Ans.  7  or  3. 

10.  Two  persons,  A  and  B,  departed  from  different  places 
at  the  same  time,  and  travelled  towards  each  other.     On 
meeting,  it  appeared  that  A  had  travelled   18  miles  more 
than  B  ;  and  that  A  could  have  gone   B's  journey  in  1 5| 
days,  but.  B  would  have  been  28  days  in  performing  A's 
journey      How  far  did  each  travel  ? 

(  A  72  miles 
Ans.    ^ 

B  54  miles 


EQUATIONS    OF    THE    SECONI*    DEGREE.  215 

11.  There  are  two  numbers  whose  difference  is  15,  cind 
half  the'ir  product  is  equal  to  the  cube  of  the  lesser  num- 

'ber.      What  are  those  numbers  1  Ans.  3  and  18. 

12.  What  two  numbers  are  those  whose  sum,  multiplied 
by  the  greater,  is  equal  to  77  ;  and  whose  difference,  multi 
plied  by  the  lesser,  is  equal  to  12  ? 

Ans.  4  and  7,  or  f  yTand  y  -/~2. 

13.  To  divide  100  into  two  such  parts,  that  the  sum  01 
their  square  roots  may  be  14.  Ans.  64  and  36. 

14.  It  is  required  to  divide  the  number  24  into  two  such 
parts,  that  their  product  may  be  equal  to  35  times  their  dif- 
ference. Ans.  10  and  14. 

15.  The  sum  of  two  numbers  is  8,  and  the  sum  of  their 
cubes  152.     What  are  the  numbers  1  Ans.  3  and  5. 

16.  Two  merchants  each  sold  the  same  kind  of  stuff; 
the  second  sold  3  yards  more  of  it  than  the  first,  and  to- 
gether they  receive  35  dollars.     The  first  said  to  the  second, 
"1   would  have  received  24 'dollars  for  your  stuff;"   the 
other  replied,  "  And  I  should  have  received  12 J  dollars  for 
yours."     How  many  yards  did  each  of  them  sell  ? 

(  1st  merchant  x  =15  x=5. 

Ans.     ?  or 

I  2nd        „'     y=18  y=8. 

17.  A  widow  possessed  13,000  dollars,  which  she  divided 
into  two  parts,  and  placed  them  at  interest,  in  such  a  man- 
ner, that  the  incomes  from  them  were  equal.     If  she  had 
put  out  the  first  portion  at  the  same  rate  as  the  second,  she 
would  have  drawn  for  this  part  360  dollars  interest ;  and  if 
she  had  placed  the  second  out  at  the  same  rate  as  the  first, 
she  would  have  drawn  for  it  490  dollars  interest.     What 
were  the  two  rates  of  interest  ?  " 

Ans.  7  and  0  per  cen 


216  ELEMENTARY  ALGEBRA 


CHAPTER  VII. 
Of  Proportions  and  Progressions. 

135.  Two  quantities  of  the  same  kind  may  be  compared 
together  in  two  ways  : — 

1st.  By  considering  how  muck  one  is  greater  or  less  than 
the  other,  which  is  shown  by  their  difference ;  and, 

2nd.  By  considering  how  many  times  one  is  greater  or 
less  than  the  other,  which  is  shown  by  their  quotient. 

Thus,  in  comparing  the  numbers  3  and  12  together  with 
respect  to  their  difference,  we  find  that  1 2  exceeds  3  by  9 ; 
and  in  comparing  them  together  with  respect  to  their  quo- 
tient, we  find  that  12  contains  3  four  times,  or  that  12  is  4 
times  as  great  as  3. 

The  first  of  these  methods  of  comparison  is  called  Arith- 
metical Proportion,  and  the  second  Geometrical  Proportion. 

Hence,  Arithmetical  Proportion  considers  the  relation  of 
quantities  with  respect  to  their  difference,  and  Geometrical 
Proportion  the  relation  of  quantities  with  respect  to  their 
quotient. 


QUKST. — 135.  In  how  many  ways  may  two  quantities  be  compared 
together  1  What  does  the  first  method  consider  ]  What  the  second  ! 
What  is  the  first  of  these  methods  called  1  What  is  the  second  called  * 
How  then  do  you  defmo  the  two  proportions'? 


ARITHMETICAL    PROPORTION.  217 


Of  Arithmetical  Proportion  and  Progression. 

136.  If  we  have  four  numbers,  2,  4,  8,  and  10,  of 

which  the  difference  between  the  first  and  second  is  equal 
to  the  difference  between  the  third  and  fourth,  these  num 
bers  are  said  to  be  in  arithmetical  proportion.  The  first 
term  2  is  called  an  antecedent,  and  the  second  term  4,  with 
which  it  is  compared,  a  consequent.  The  number  8  is  also 
called  an  antecedent,  and  the  number  10,  with  which  it  is 
compared,  a  consequent. 

When  the  difference  between  the  first  and  second  is  equal 
to  the  difference  between  the  third  and  fourth,  the  four  num- 
bers are  said  to  be  in  proportion.  Thus,  the  numbers 

2,     4,     8,     10, 
are  in  arithmetical  proportion. 

137.  When  the  difference  between  the  first  antecedent 
and  consequent  is  the  same  as  between  any  t\vo  adjacent 
terms  of  the  proportion,  the  proportion  is  called  an  arith- 
metical progression.     Hence,  a  progression  by  differences,  or 
an  arithmetical  progression,  is  a  series  in  which  the  succes- 
sive terms  continually  increase  or  decrease  by  a  constant 
number,   which   is    called   the    common   difference   of    the 
progression. 

Thus,  in  the  two  series 

1,     4,     7,  10,  13,  16,  19,  22,  25,  ... 
60,  56,  52,  48,  44,  40,  36,  32,  28,  ... 


QUEST. — 136.  When  arc  four  numbers  in  arithmetical  proportion? 
What  is  the  first  called  1  What  is  the  second  called  1  What  is  the 
third  called  1  What  is  the  fourth  called  ? 


218  ELEMENTARY  ALGEBRA. 

the  first  is  called  an  increasing  progression,  of  which  trie 
common  difference  is  3,  and  the  second  a  decreasing  pro- 
gression, of  which  the  common  difference  is  4. 

In  general,  let  0,  &,  c,  e?,  e,f,  .  .  .  designate  the  terms 
of  a  progression  by  differences  ;  it  has  been  agreed  to  write 
them  thus : 

a.b.c.d.e.f.g.h.i.k... 

This  series  is  read,  a  is  to  b,  as  b  is  to  c,  as  c  is  to  d,  as  d 
is  to  e,  &c.  This  is  a  series  of  continued  equi-differences, 
in  which  each  term  is  at  the  same  time  a  consequent  and 
antecedent,  with  the  exception  of  the  first  term,  which  is 
only  an  antecedent,  and  the  last,  which  is  only  a  consequent. 

138.  Let  r  represent  the  common  difference  of  the 
progression 

a.b.c.d.e.f.g.h,  &c, 

which  we  will  consider  increasing. 

From  the  definition  of  the  progression,  it  evidently  follows 
that 

b=a+r,     c=b  +  r=a+2r,     d=c+r=a+3r ; 

and,  in  general,  any  term  of  the  series  is  equal  to  the  first 
term  plus  as  many  times  the  common  difference  as  there  are 
preceding  terms. 

Thus,  let  /  be  any  term,  and  n  the  number  which  marks 
the  place  of  it :  the  expression  for  this  general  term  is 

l=a+(n  —  l)r. 


QUEST. — 137.  What  is  an  arithmetical  progression  1  What  is  the 
number  called  by  which  the  terms  are  increased  or  diminished  1  What 
13  an  increasing  progression  1  What  is  a  decreasing  progression  I 
Which  term  is  only  an  antecedent  1  Which  only  a  consequent 7 


ARITHMETICAL    PROGRESSION.  219 

Hence,  for  finding  the  last  term,  we  have  the  following 

RULE. 

I.  Multiply  the  common  difference  by  one  less  than  the 
number  of  terms. 

II.  To  the  product  add  the  fast  term:  the  sum  will  be  the 
last  term. 

EXAMPLES. 

The  formula  l=a-\-(n — l)r  serves  to  find  any  term 
whatever,  without  our  being  obliged  to  determine  ail  those 
which  precede  it. 

1.  If  we  make  n=l,  we  have  l=<!,,  that  is,  the  series 
will  have  but  one  term. 

2.  If  we  make  n=2,  we  have  l=a-*-r  ;  that  is,  the  series 
will  have  two  terms,  and  the  second  term  is  equal  to  the 
first  plus  the  common  difference. 

3.  If  a=3  and  r=2,  what  is  the  3rd  term  ?  Ans.  7. 

4.  If  a=5  and  r=4,  what  is  the  6th  term?  Ans.  25. 

5.  If  a=*  and  r=5,  what  is  the  9th  term?  Ans.  47. 

6.  If  a=8  and  r=5,  what  is  the  10th  term  ? 

Ans.  53. 
1  If  a  =  20  and  r=4,  what  is  the  12th  term  ? 

Ans.  64. 
8.  If  a=40  and  r=20,  what  is  the  50th  term  ? 

Ans.   1020. 


QUEST. — 138.  Give  the  rule  for  finding  the  last  term  of  a  series  when 
the  progression  is  increasing. 


220  ELEMENTARY  ALGEBRA. 

9    If  0=45  and  r= 30,  what  is  the  40th  term  ? 

Ans.  1215 

10.  If  a=30  and  r=20,  what  is  the  60th  term? 

Ans.  1210 

11.  If  a  =  50  and  r=10,  what  is  the  100th  term? 

Ans.  1040 

12.  To  find  the  50th  term  of  the  progression 

1   .  4  .  7  .  10  .  13  .  16  .  19  .  .  ., 
we  have  2=1+49x3=148. 

13.  To  find  the  60th  term  of  the  progression 

1   .  5  .  9  .  13  .  17  .  21   .  25  .  .  ., 
we  have  1=  1+59x4 =237. 

139.  If  the  progression  were  a  decreasing  one,  we 
bhould  have 

l=a— (n— I)r. 

Hence,  to  find  the  last  term  of  a  decreasing  progression, 
we  have  the  following 

RULE. 

I.  Multiply  the  common  difference  by  one  less  than  the  num- 
ber of  terms. 

II.  Subtract  the  product  from  the  first  term  .   the  remaindei 
will  be  the  last  term 


QUEST. — 1"39.  Give  the  rule  for  finding  the  last  term  of  a  series, 
when  the  progression  is  decreasing. 


ARITHMETICAL    PROGRESSION.  221 


EXAMPLES. 

1.  The  first  term  of  a  decreasing  progression  is  60,  the 
number  of  terms  20,  and  the  common  difference  3  :  what 
is  tne  last  term  ? 

l-a-(n-l)r     gives     1=60  —  (20  — 1)3  =  60  —  57  =  3. 

2.  The  first  term  is  90,  the  common  difference  4,  and 
the  number  of  terms  1 5  :  what  is  the  last  term  1     Ans.  34. 

3.  The  first  term  is  100,  the  number  of  terms  40,  and  the 
common  difference  2  :  what  is  the  last  term  1         Ans.  22. 

4.  The  first  term  is  80,  the  number  of  terms  10,  and  the 
common  difference  4  :  what  is  the  last  term  1          Ans.  44. 

5.  The  first  term  is  600,  the  numl^r  of  terms  100,  and 
the  common  difference  5  :  what  is  the  last  term  ? 

Ans.  105. 

6.  The  first  term  is  800,  the  number  of  terms  200,  and 
the  common  difference  2  :   what  is  the  last  term  ? 

Ans.  402. 

1 4O.  A  progression  by  differences  being  given,  it  is 
proposed  to  prove  that,  the  sum  of  any  two  terms,  taken  at 
equal  distances  from  the  two  extremes,  is  equal  to  the  sum  of 
the  two  extremes. 

That  is,  if  we  have  the  progression 

2  .  4  .  6  .  8  .  10  .  12, 

we  wish  to  prove  that 

4+10     or     6  +  8 

is  equal  to  the  sum  of  the  two  extremes  2  and  12. 

20 


222  ELEMENTARY  ALGEBRA. 

Let  a.i>.c.d.c.j....i.k.l  be  the  pro- 
posed progression,  and  n  the  number  of  terms. 

We  will  first  observe  that,  if  x  denotes  a  term  which  has 
p  terms  before  it,  and  y  a  term  which  has  p  terms  after  it 
we  have,  from  what  has  been  said, 

x=a+pxr, 

and  y=l—pxr\ 

whence,  by  addition,      x-}-y=a-}-l. 

Which  demonstrates  the  proposition. 

Referring  this  proof  to  the  previous  example,  if  we  sup- 
pose, in  the  first  place,  x  to  denote  the  second  term  4,  then 
y  will  denote  the  term  10,  next  to  the  last.  If  x  denotes 
the  3rd  term  6,  thei*  y  will  denote  8,  the  third  term  from 
the  last. 

Having  proved  the  first  part  of  the  proposition,  write  the 
progression  below  itself,  but  in  an  inverse  order,  viz  : 

a.b.c.d.e.f...i.k.l. 
I  .  k  .  i  .........  c  .  b  .  a. 

Calling  S  the  sum  of  the  terms  of  the  first  progression, 
2S  will  be  the  sum  of  the  terms  in  both  progressions,  and 
we  shall  have 


Now,  since  all  the  parts  a-\~l,  b-\-k,  c-\-i  .  .  .  are  equal 
to  each  other,  and  their  number  equal  to  n, 


or        ±= 


ARITHMETICAL    PROGRESSION.  223 

Hence,  for  finding  the  sum  of  an  arithmetical  series,  we 
have  the  following 


RULE. 

I.  Add  the  two  extremes  together,  and  take  half  their  s'<im. 

II.  Multiply   the  half-sum  by   the   number  of  terms ;    the 
product  will  be  the  sum  of  the  series. 

EXAMPLES. 

I.  The  extremes  are  2  and  16,  and  the  number  of  terms 
8  :  what  is  the  sum  of  the  series  ? 

S=( )X«,     gives     Sr=- 

3.  The  extremes  are  3  and  27,  and  the  number  of  terms 
12  :  what  is  the  sum  of  the  series  ?  Ans.  180. 

3.  The  extremes  are  4  and  20,  and  the  number  of  terms 
10  :  what  is  the  sum  of  the  series  ?  Ans.  120. 

4.  The  extremes  are   100  and  200,  and  the  number  of 
terms  80  :  what  is  the  sum  of  the  series  ?         Ans.  12000. 

5.  The  extremes  are  500  and  60,  and  the  number  of  terms 
20  :  what  is  the  sum  of  the  series  ?  Ans.  5600. 

6.  The  extremes  are  800  and  1200,  and  the  number  of 
terms  50  :  what  is  the  sum  of  the  series  ?         Ans.  50000. 


QUEST. — 140.  In  every  progression,  what  is  the  sum  of  the  twc  ex- 
tremes equal  to  1  What  is  the  rule  for  finding  the  sum  of  an  arithmeti- 
cal scries  ? 


224  ELEMENTARY  ALGBBRA. 

141.  In  arithmetical  proportion  there  are  five  number 
to  be  considered  :  — 

1st.  The  first  term,  a. 
2nd.  The  common  difference,  r. 
3rd.  The  number  of  terms,  n. 
4th.  The  last  term,  I. 

5th.  The  sum,  S. 

\ 

The  formulas 

J=a+n-l)r     and     S= 


contain  five  quantities,  a,  r,  n,  I,  and  S,  arid  consequently 
give  rise  to  the  following  general  problem,  viz  :  Any  three 
of  these  Jive  quantities  being  given,  to  determine  the  other 
two. 

We  already  know  the  value  of  *S  in  terms  of  a,  n,  and  7 
From  the  formula 

J=a+(n-l)r, 
we  find  a  =  l—  (n—  l)r. 

That  is  :    The  first  term  of  an  increasing  arithmetical  pro- 
gression is  equal  to  the  last  term,  minus  the  product  of  the 
common  difference  by  the  number  of  terms  less  one. 
From  the  same  formula,  we  also  find 

I—  a 


n  —  l 


That  is  :  In  any  arithmetical  progression,  the  common  differ- 
ence is  equal  to  the  difference  between  the  two  extremes  divided 
by  the  number  of  terms  less  one. 

QUEST. — 141.  How  many  numbers  are  considered  in  arithmetical 
proportion  1  What  are  they  1  In  every  arithmetical  progression,  what 
is  the  common  difference  equal  to  1 


ARITHMETICAL    PROGRESSION.  225 

The  last  term  is  16,  the  first  term  4,  and  the  number  of 
terms  5  :  what  is  the  common  difference  ? 

The  formula  r= 


n  —  1 
16-4 


r= 


2.  The  last  term  is  22,  the  first  term  4,  and  the  number 
of  terms  10  :  what  is  the  common  difference  ?  Ans.  2. 

1  42.  The  last  principle  affords  a  solution  to  the  follow- 
ing question  : 

To  find  a  number  m  of  arithmetical  means  between  two 
given  numbers  a  and  b. 

To  resolve  this  question,  it  is  first  necessary  to  find  the 
common  difference.  Now,  we  may  regard  a  as  the  first 
term  of  an  arithmetical  progression,  b  as  the  last  term,  and 
the  required  means  as  intermediate  terms.  The  number  of 
terms  of  this  progression  will  be  expressed  by  w-j-2. 

Now,  by  substituting  in  the  above  formula,  b  for  I,  and 
Tn-f-2  for  n,  it  becomes 


m  +  2  —  1 

that  is,  the  common  difference  of  the  required  progression  is 
obtained  by  dividing  the  difference  between  the  given  num- 
bers a  and  &,  by  one  more  than  the  required  number  of 
means. 


QUEST. — 142.   How  do  you  find  any  number  of  arithmetical  means 
oetween  two  g'iven  numbers  1 

20* 


226  ELEMENTARY  ALGEBRA. 

Having  obtained  the  common  difference,  form  the  second 
term  of  the  progression,  or  the  first  arithmetical  mean,  by 

adding  r,  or   -,  to  the  first  term  a.     The  second  mean 

m-\- 1 

;s  obtained  by  augmenting  the  first  by  r,  &c. 

1.  Find  three  arithmetical  means  between  the  extremes 
2  and  18. 

The  formula  r= 


18-2 
gives  r=  — - — =4  ; 

hence,  the  progression  is 

2  .  6  .   10  .   14  .   18. 

2.  Find  twelve  arithmetical  means  between  12  and  7V. 

b-a 
The  formula 

gives  r=^-^-=5. 

Hence  the  progression  is 

12  .  17  .  22  .  27  ....  77 

143.  REMARK.  If  the  same  number  of  arithmetical 
means  are  inserted  between  all  the  terms,  taken  two  and 
two,  these  terms,  and  the  arithmetical  means  united,  will 
form  but  one  and  the  same  progression. 

For,  let  a.b.c.d.e.f...  be  the  proposed 
progression,  and  m  the  number  of  means  to  be  inserted 
between  a  and  b,  b  and  c,  c  and  d  .  .  . 


ARITHMETICAL    PROGRESSION.  227 

From  what  lias  just  been  said,  the  common  difference  of 
each  partial  progression  will  be  expressed  by 


which  are  equal  to  each  other,  since  a,  5,  c  ...  are  in 
progression  :  therefore,  the  common  difference  is  the  same 
in  each  of  the  partial  progressions  ;  and  since  the  last  term 
of  the  first,  forms  the  first  term  of  the  second,  &c,  we  may 
conclude  that  all  of  these  partial  progressions  form  a  single 
progression. 


EXAMPLES. 

1.  Find  the  sum  of  the  first  fifty  terms  of  the  progression 
2  .  9  .   16  .  23  ... 

For  the  50th  term  we  have 


Hence,        S  =  2  +  345  X      =  347x25  =  8675. 


2.  Find  the  100th  term  of  the  series  2  .  9  .  16  .  23  ... 

Ans.  695 

3.  Find  the  sum  of  100  terms  of  the  series  1  .  3_  5  . 
7.9  ...  Ans.   ]  0000 

4.  The  greatest  term  is  70,  the  common  difference  3,  and 
the  number  of  terms  21  :    what  is  the  least  term  and  the 
sum  of  the  series  ? 

Ans.  Least  term  10  ;  sum  of  serifs  840 


228  ELEMENTARY  ALGEBRA. 

5  The  first  term  is  4,  the  common  difference  8,  and  the 
number  of  terms  8  :  what  is  the  last  term,  and  the  sum  of 
the  series  ? 

Last  term  60. 

Sum     =  256. 

6.  The  first  term  is  2,  the  last  term  20,  and  the  number 
of  terms  10  :  what  is  the  common  difference  ? 

Ans.  2. 

7.  Insert  four  means  between  the  two  numbers  4  and  1 9  : 
what  is  the  series  ? 

Ans.    4  .  7  .  10  :   13  .  16  .  19. 


sion  is  10,  the  common  difference  — ,  and  the  number  of 

o 


8.  The  first  term  of  a  decreasing  arithmetical  progres- 
sn  is  10,  the  common  difference  — ,  and 

o 

terms  21  :  required  the  sum  of  the  series. 

Ans.   140. 

9.  In   a  progression  by   differences,  having   given  the 
common  difference  6,  the  last  term  185,  and  the  sum  of  the 
terms  2945  :  find  the  first  term,  and  the  number  of  terms. 

Ans.  First  term  =5  ;   number  of  terms  31. 

10.  Find  nine  arithmetical  means  between  each  antece- 
dent and  consequent  of  the  progression  2.5.8.11.14  ... 

Ans.   Common  dif.,  or  n=0,3. 

11.  Find  the  number  of  men  contained  in  a  triangular  bat- 
talion, the  first  rank  containing  one  man,  the  second  2,  the 
third  3,  and  so  on  to  the  n*,  which  contains  n.     In  other 
words,  find  the  expression  for  the  sum  of  the  natural  num- 
bers 1,  2,  3  .  .  .,  from  1  to  n  inclusively. 

A*..  S= 


GEOMETRICAL    PROPORTION.  229 

12.  Find  the  sum  of  the  n  first  terms  of  the  progression 
of  uneven  numbers   1,  3,  5,  7,  9  ...  Ans.  S  —  nz. 

13.  One  hundred  stones  being  placed  on  the  ground  in  a 
straight  line,  at  the  distance  of  2  yards  from  each  other, 
.how  far  will  a  person  travel  who  shall  bring  them  one  by 
One  to  a  basket,  placed  at  2  yards  from  the  first  stone  ? 

Ans.   11  miles,  840  yards 


Geometrical  Proportion  and  Progression. 

144.  Ratio  is  the  quotient  arising  from  dividing  one 
quantity  by  another  quantity  of  the  same  kind.  Thus,  if 
the  numbers  3  and  6  have  the  same  unit,  the  ratio  of  3  to  6 
will  be  expressed  by 


And  in  general,  if  A  and  B  represent  quantities  of  the  same 
kind,  the  ratio  of  A  to  B  will  be  expressed  by 

B_ 
A' 

145.  If  there  be  four  numbers 

2,     4,     8,     16, 

having  such  values  that  the  second  divided  by  the  first  is 
equal  to  the  fourth  divided  by  the  third,  the  numbers  are 


QUEST.— 144.  What  is  ratio  1     What  is  the  ratio  of  3  to  6?     Of  4 
to  12? 


230  ELEMENTARY  ALGEBRA. 

.said  to  be  in  proportion.     And  in  general,  if  there  be  foui 
quantities,  A,  B,  C,  and  D,  having  such  values  that 

B      D 


then  A  is  said  to  have  the  same  ratio  to  B  that  C  has  to  D 
or,  the  ratio  of  A  to  B  is  equal  to  the  ratio  of  C  to  D, 
When  four  quantities  have  this  relation  to  each  other,  they 
are  said  to  be  in  proportion.     Hence,  proportion  is  an  equality. 
of  ratios. 

To  express  that  the  ratio  of  A  to  B  is  equal  to  the  ratio 
of  C  to  Z),  we  write  the  quantities  thus  : 


and  read,  A  is  to  B  as  C  to  D. 

The  quantities  which  are  compared  together  are  called 
the  terms  of  the  proportion.  The  first  and  last  terms  are 
called  the  two  extremes,  and  the  second  and  third  terms,  the 
two  means.  Thus,  A  and  D  are  the  extremes,  and  B  and 
C  the  means. 

1  46.  Of  four  proportional  quantities,  the  first  and  third 
are  called  the  antecedents,  and  the  second  and  fourth  the 
consequents  ;  and  the  last  is  said  to  be  a  fourth  proportional 
to  the  other  three  taken  in  order.  Thus,  in  the  last  pro- 
portion A  and  C  are  the  antecedents,  and  B  and  D  the 
consequents. 


QUEST. — 145.  What  is  proportion  1  How  do  you  express  that  foui 
numbers  are  in  proportion  1  What  are  the  numbers  called  1  What  arc 
the  first  and  fourth  called?  What  the  second  and  third! — 146.  In  four 
proportional  quantities,  what  are  the  first  and  third  called  ?  What  the 
second  and  fourth  1 


GEOMETRICAL    PROPORTION.  231 

1 47 .  Three  quantities  are  in  proportion  when  the  first 
has  the  same  ratio  to  the  second  that  the  second  has  to  the 
third ;  and  then  the  middle  term  is  said  to  be  a  mean  pro- 
portional between  the  other  two.     For  example, 

3  :  6  :  :  6  :   12; 
and  6  is  a  mean  proportional  between  3  and  12. 

148.  Quantities  are  said  to  be  in  proportion  by  inver- 
sion, or  inversely,  when  the  consequents  are  made  the  ante- 
cedents and  the  antecedents  the  consequents. 

Thus,  if  we  have  the  proportion 

3  :  6  :  :  8  :   16, 
the  inverse  proportion  would  be 

6  :  3  :  :   16  :  8. 

1 49.  Quantities  are  said  to  be  in  proportion  by  alterna- 
tion, or  alternately,  when  antecedent  is  compared  with  ante- 
cedent and  consequent  with  consequent. 

Thus,  if  we  have  the  proportion 

3  :  6  :  :  8  :  16, 
the  alternate  proportion  would  be 

3  :  8  :  :  6  :   16. 


QUEST. — 147.  When  are  three  quantities  proportionall  What  is  the 
middle  one  called? — 148.  When  are  quantities  said  to  be  in  proportion 
by  inversion,  or  inversely  1 — 149.  When  are  quantities  in  proportion  by 
alternation  * 


232  ELEMENTARY  ALGEBRA. 

150.  Quantities  are  said  to  be  in  proportion  by  compo- 
sition, when  the  sum  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

2  :  4  :  :  8  .   16, 
die  proportion  by  composition  would  be 

2  +  4  :  4  :  :   8  +  16  :   16; 
that  is,  6:4::         24  :   16. 

151.  Quantities  are  said  to  be  in  proportion  by  division, 
when  the  difference  of  the  antecedent  and  consequent  is 
compared  either  with  antecedent  or  consequent. 

Thus,  if  we  have  the  proportion 

3  :  9  :  :   12  :  36, 
the  proportion  by  division  will  be 

9  —  3  :  9  :  :  36-12  :   36; 
that  is,  6  :  9  :  :  24  :  36. 

152.  Equi-multiples  of  two  or  more  quantities  are  the 
products  which  arise  from  multiplying  the  quantities  by  the 
same  number. 

Thus,  if  we  have  any  two  numbers,  as  6  and  5,  and  mul- 
tiply them  both  by  any  number,  as  9,  the  equi-multiples  will 
be  54  and  45  ;  for 

6x9  =  54,     and     5x9  =  45. 


QUE8T. — 150.  When  are  quantities  in  proportion  by  composition? 
— 151.  When  are  quantities  in  proportion  by  division"! — 152.  What 
are  equi-multiples  of  two  or  more  quantities  1 


GEOMETRICAL    PROPORTION.  233 

Also,  m X  A  and  mxB  are  equi-multiples  of  A  and  B,  the 
common  multiplier  being  m. 

153.  Two  quantities,  A  and  B,  are  said  to  be  recipro- 
cally proportional,  or  inversely  proportional,  when  one   in- 
creases in  the  same  ratio  as  the  other  diminishes.     When 
this  relation  exists,  either  of  them  is  equal  to  a  constant 
quantity  divided  by  the  other. 

Thus,  if  we  had  any  two  numbers,  as  2  and  4,  so  related 
to  each  other  that  if  we  divided  one  by  any  number  we  must 
multiply  the  other  by  the  same  number,  one  wduld  increase 
just  as  fast  as  the  other  would  diminish,  and  their  product 
would  be  constant. 

154.  If  we  have  the  proportion 

A  :  B  :  :   C  :  D, 

7?      D 
we  have  ~A=~C'  (Art  14^) ; 

and  by  clearing  the  equation  of  fractions,  we  have 
BC=AD. 

That  is,  Of  four  proportional  quantities,  the  product  of  the 
two  extremes  is  equal  to  the  product  of  the  two  means. 

This  general  principle  is  apparent  in  the  proportion  be- 
tween the  numbers 

2  :   10  :  :   12  :  60, 
which  gives  2x60  =  10x12  =  120. 


QUEST. — 153.  When  are  two  quantities  said  to  be  reciprocally  pro- 
portional 1 — 154.  If  four  quantities  are  proportional,  what  is  the  product 
of  the  two  means  equal  to  ? 


2,'M  ELEMENTARY  ALGEBRA. 

1  55.  If  four  quantities,  A,  B,  C,  D,  are  so  related**- 
each  other  that 

AxD=BxCt 

we  shall  also  have  -r=~77> 

A      C 

and  hence,  A  :  B  :  :   C  :  D. 

That  is :  If  the  product  of  two  quantities  is  equal  to  the.  pro- 
duct of  two  other  quantities,  two  of  them  may  be  made  the 
extremes,  and  the  other  two  the  means  of  a  proportion, 
Thus,  if  we  have 

2x8=4x4, 
we  also  have 

2  :  4  :  :  4  :  8. 

156.  If  we  have  three  proportional  quantities 
A  :  B  :  :  B  :   C, 

B      C 
we  have  — = -^- ; 

hence,  B2=AC. 

That  is  :    The  square  of  the  middle  term  is  equal  to  the  product 
of  the  two  extremes. 

Thus,  if  we  have  the  proportion 

3  :  6  : :  6  :  12, 

we  shall  also  have  / 


QUEST. — 155.  If  the  product  of  two  quantities  is  equal  to  the  product 
of  two  other  quantities,  may  the  four  be  placed  in  a  proportion  1  How  1 
— 156.  If  three  quantities  are  proportional,  what  is  the  product  of  thf 
extremes  equal  to1 


GEOMETRICAL    PROPORTION.  235 

157.  If  we  have 

B     D 

A  :  B  :  :   C  :  D,    and  consequently    -— =TT, 

A       \j 

c 

multiply  both   members  of  the  last  equation   by  -=-,   we 
then  obtain, 


and  hence,  A  :   C  :  :  B  :  D. 

That  is  :    If  four  quantities  are  proportional,  they  will  be  in 
proportion  by  alternation. 

Let  us  take,  as  an  example, 

10  :   15  :  :  20  :  30. 
We  shall  have,  by  alternating  the  terms, 

10  :  20  :  :   15  :  30. 
158.  If  we  have 

A  :  B  :  :   C  :  D    and    A  :  B  :  :  E  :  F, 
we  shall  also  have 

B _D  B  _F 

~A~~C  ~A~~E' 

D      F 

hence,  -7^=-^   and   C  :  D  :  :  E  :  F. 

L>        TJ 

That  is  :    If  there  are  two  sets  of  proportions  having  an 


QUEST. — 157.  If  four  quantities  are  proportional,  will  they  be  in  pro- 
portion by  alternation  1 


236  ELEMENTARY  ALGEBRA. 

antecedent  and  consequent  in  the  one  equal  to  an  antecedent 
and  consequent  of  the  other ',  the  remaining  terms  will  be  pro 
portional. 

If  we  have  the  two  proportions 

2  :  6  :  :  8  :  24     and     2  :  6  :  :  10  :  30, 
we  shall  also  have 

8  :  24  :  :   10  :  30. 
159.  If  we  have 

TO  T\ 

A  :  B  :  :   C  :  D,    and  consequently    — =-— , 

we  have,  by  dividing  1  by  each  member  of  the  equation 

A      C 

—  =-pr,   and  consequently   B  :  A  :  :  D  :   C. 

Jj       D 

That  is  :  Four  proportional  quantities  will  be  in  proportion^ 
when  taken  inversely. 

To  give  an  example  in  numbers,  take  the  proportion 

7  :   14  :  :   8  :   16; 
then,  the  inverse  proportion  will  be 

14  :  7  :  :   16  :  8, 
in  which  the  ratio  is  one-half. 
1  GO.  The  proportion 

A  :  B  ::   C  :  D     gives     AxD=BxC. 


QUEST. — 158.  If  you  have,  two  sets  of  proportions  having  an  ante- 
cedent and  consequent  in  each,  equal ;  what  will  follow  1 — 159.  If  four 
quantities  are  in  proportion,  will  they  be  in  proportion  when  takei  in- 
versely 1 


GEOMETRICAL    PROPORTION.  237 

To  each  member  of  the  last  equation   add  BxD.     We 
shall  then  have 


and  by  separating  the  factors,  we  obtain 

A+B  :  B  :  :   C+D  :  D. 

If,  instead  of  adding,  we  subtract  B  x  D  from  both  mem- 
bers, we  have 


which  gives 

A-B  :  B  :  :   C-D  :  D. 

That  is  :    If  four  quantities  are  proportional,  they  will  be  tn 
proportion  by  composition  or  division. 

Thus,  if  we  have  the  proportion 

9  :  27  :  :  16  :  48, 
we  shall  have,  by  composition, 

9  +  27  :  27  :  :   16  +  48  :  48: 
that  is,  36  :  27  :  :  64  :  48, 

in  which  the  ratio  is  three-fourths. 
The  proportion  gives  us,  by  division, 

27  —  9  :  27  :  :  48  —  16  :  48; 
that  is,  18  :  27  :  :  32  :  48, 

in  which  the  ratio  is  one  and  one  -half. 


QUEST. — 160.  If  four  quantities  are  in  proportion,  will  they  he  in  rn>- 
portion  by  composition  1     Will  they  be  in  proportion  by  division  1     What 
>«  the  fWFerence  between  composition  and  division  1 
21* 


238  ELEMENTARY  ALGEBRA. 

161.  If  we  have 

B_D 
~A~~C* 

and  multiply  the  numerator  and  denominator  of  the  fiist 
member  by  any  number  m,  we  obtain 

v    and   mA  .  mB  ..  c  .  Dm 


mA       C 

That  is  :  Equal  multiples  of  two  quantities  have  the  same 
ratio  as  the  quantities  themselves. 

For  example,  if  we  have  the  proportion 

5  :   10  :  :   12  :  24, 

and  multiply  the  first  antecedent  and  consequent  by  6,  we 
have 

30  :  60  :  :   12  :  24, 

in  which  the  ratio  is  still  2. 
162.  The  proportions 

A  :  B  :  :    C  :  D     and     A  :  B  :  :  E  :  F, 
give          AxD=BxC     and     AxF=BxE-, 
adding  and  subtracting  these  equations,  we  obtain 
A(D±F)  =  B(C±E),     or     A  :  B  ::   C±E  :  D±F 

That  is  :  If  C  and  D,  the  antecedent  and  consequent,  be  aug- 
mented or  diminished  by  quantities  E  and  F,  which  have  the 
same  ratio  as  C  to  D,  the  resulting  quantities  will  also  have 
the  same  ratio. 


QUEST. — 161.  Have  equal  multiples  of  two  quantities  the  same  ratii 
as  the  quantities'? — 162.  Suppose  the  antecedent  and  consequent  bi 
augmented  or  diminished  by  quantities  having  the  same  ratio  .' 


GEOMETRICAL    PROPORTION.  239 

Let  us  take,  as  an  example,  the  proportion 
9  :   18  :  :  20  :  40, 

b    which  the  ratio  is  2. 

If  we  augment  the  antecedent  and  consequent  by  15  and 
3U  vrhich  have  the  same  ratio,  we  shall  have 

9  +  15  :   18  +  30  :  :  20  :  40; 
that  is,  24  :  48  :  :  20  :  40, 

in  which  the  ratio  is  still  2. 

If  we  diminish  the  second  antecedent  and  consequent  by 
the  same  numbers,  we  have 

9  :   18  :  :  20  —  15  :  40—30; 
that  is,  9  :   18  :  :  5  :   10, 

in  wLich  the  ratio  is  still  2. 

163.  If  we  have  several  proportions 

A  :  B  ::  C  :  D,  which  gives  Axi*  =  BxC, 
A  :  B  :  :  E  :  F,  „  „  AxF-BxE, 
A  :  B  :  :  G  :  H,  „  „  AxH=B*G. 

&c,  &c, 
we  shJfl  have,  by  addition, 

A(D+F+H)=B(C+E+G); 
and  by  separating  the  factors, 

A  :   B  :  :   C+E+G  :  D+F+H. 

That  is  :  In  any  number  of  proportions  having  the  same 
ratio,  any  antecedent  will  be  to  its  consequent,  as  the  sum  uj 
the  antecedents  to  the  sum  of  the  consequents. 


y-'  ELEMENTARY  ALGEBRA. 

Let  us  take,  for  example, 

2  :  4  :  :  6  :   12     and     1   :  2  :  :  3  :  6,     &c 
Then,  2:4::  6+3  :   12  +  6; 

that  is,  2  :  4  :  :  9  :   18, 

iu  which  the  ratio  is  still  2. 

164.  If  we  have  four  proportional  quantities 

A  :  B  :  :   C  :  D,     we  have     ^=^  ; 
and  raising  both  members  to  any  power,  as  »,  we  have 


and  consequently 

A*  :  Bn  :  :   C*  :  D\ 

That  is  :    If  four  quantities  arc  proportional,  any  like  powers 
or  roots  will  be  proportional. 

If  we  have,  for  example, 

2  :  4  :  :  3  :  6,    * 
we  shall  have         22   :   42  :  :  32  :  62  ; 
that  is,  4   :   16   :  :   9  :  36, 

in  which  the  terms  are  proportional,  the  ratio  being  4. 

165.  Let  there  be  two  sets  of  proportions, 

7?      7} 
A  :  B  :  :   C  :  D,    which  gives     —  =—, 

E  :  F  :  :   G  :  //,         „         „         ~=~' 


QUEST. — 163.  In  any  number  of  proportions  having  the  same  rati 
how  will  any  one  antecpdent  be  to  its  consequent  1 — 164.  In  four  pr<. 
portional  quantities,  how  are  like  powers  or  roots'? 


GEOMETRICAL    PROGRESSION.  241 

Multiply  them  together,  member  by  member,  we  have 
BF      DH 


AE~  CG 


which  gives     AE  :  BF  :  :   CG  :  DH. 


That  is  :    In  two  sets  of  proportional  quantities,  the  products 
of  the  corresponding  terms  will  be  proportional. 

Thus,  if  we  have  the  two  proportions 

8  :   16  :  :   10  :     20 
and  3  :     4  :  :     6  :       8, 

we  shall  have        24  :  64  :  :  60  :   160. 

Geometrical  Progression. 

166.  We  have  thus  far  only  required  that  the  ratio  of 
the  first  term  to  the  second  should  be  the  same  as  that  of 
the  third  to  the  fourth. 

If  we  impose  the  farther  condition,  that  the  ratio  of  the 
second  term  to  the  third  shall  also  be  the  same  as  that  of  the 
first  to  the  second,  or  of  the  third  to  the  fourth,  we  shall  have 
a  series  of  numbers,  each  one  of  which,  divided  by  the 
preceding  one,  will  give  the  same  ratio.  Hence,  if  any 
term  be  multiplied  by  this  quotient,  the  product  will  be  the 
succeeding  term.  A  series  of  numbers  so  formed  is  called 
a  geometrical  progression.  Hence, 

A  Geometrical  Progression,  or  progression  by  quotients,  is 
a  series  of  terms,  each  of  which  is  equal  to  the  product  of 


QUEST. — 165.  In  two  sets  of  proportions,  how  are  the  products  of  the 
corresponding  terms  7 


24x5  ELEMENTARY  ALGEBRA. 

that  which  precedes  it  by  a  constant  number,  which  number 
is  called  the  ratio  of  the  progression.  Thus, 

1   :  3  :  9  :  27  :  81   :  243,  &c, 

is  a  geometrical  progression,  which  is  written  by  merely 
placing  two  dots  between  each  two  of  the  terms.  Also, 

64  :  32  :  16  :  8  :  4  :  2  :  1 

is  a  geometrical  progression,  in  which  the  ratio  is  one-half. 

In  the  first  progression  each  term  is  contained  three  times 
in  the  one  that  follows,  and  hence  the  ratio  is  3.  In  the 
second,  each  term  is  contained  one-half  times  in  the  one 
which  follows,  and  hence  the  ratio  is  one-half. 

The  first  is  called  an  increasing  progression,  and  the 
second  a  decreasing  progression. 

Let  a,  b,  c,  d,  et  f,  .  .  .  be  numbers  in  a  progression  by 
quotients  ;  they  are  written  thus  : 

a  :  I  :  c  :  d  :  e  :  f  :  g  .  .  . 

and  it  is  enunciated  in  the  same  manner  as  a  progression  by 
differences.  It  is  necessary,  however,  to  make  the  distinc- 
tion, that  one  is  a  series  of  equal  differences,  and  the  other 
a  series  of  equal  quotients  or  ratios.  It  should  be  remarked 
that  each  term  is  at  the  same  time  an  antecedent  and  a  con- 
sequent, except  the  first,  which  is  only  an  antecedent,  and 
the  last,  which  is  only  a  consequent. 


QUEST. — 166.  What  is  a  geometrical  progression?  What  is  the  ratio 
of  the  progression'?  If  any  term  of,  a  progression  be  multiplied  by  the 
ratio,  what  will  the  product  be  1  If  any  term  be  divided  by  the  ratio, 
what  will  the  quotient  be  1  How  is  a  progression  by  quotients  written  ? 
Which  of  the  terms  is  only  an  antecedent  ?  Which  only  a  consequent T 
How  may  each  of  the  others  be  considered  1 


GEOMETRICAL    PROGRESSION.  243 

IG7.  Let  q  denote  the  ratio  of  the  progression 
a  :  b  :  c  :<?...; 

<t  being   >1    when  the  progression  is  increasing,  and  ^<  1 
when  it  is  decreasing.     Then,  since 

b__         _e__          d__         _e__ 
a  b  c  d 

we  have 


that  is,  the  second  term  is  equal  to  aq,  the  third  to  aq2,  the 
fourth  to  aq3,  the  fifth  to  a^4,  &c  ;  and  in  general,  any  term 
»,  that  is,  one  which  has  n  —  1  terms  before  it,  is  expressed 
by  aqn~l. 

Let  /  be  this  term  ;  we  then  have  the  formula 


by  means  of  which  we  can  obtain  any  term  without  being 
obliged  to  find  all  the  terms  which  precede  it.  Hence,  to 
find  the  last  term  of  a  progression,  we  have  the  following 

RULE. 

I.  Raise  the  ratio  to  a  power  whose  exponent  is  one  less  than 
the  number  of  terms. 

II.  Multiply  the  power  thus  found  by  the  first  term  :  the 
product  will  be  the  required  term. 

QUEST.  —  167.  By  what  letter  do  we  denote  the  ratio  of  the  progres- 
sion 1  In  an  increasing  progression  is  q  greater  or  less  than  1  *  In  a 
decreasing  progression  is  q  greater  or  less  than  11  If  a  is  the  firt  term 
and  q  the  ratio,  what  is  the  second  term  equal  to  1  What  the  i  ird  \ 
What  the  fourth  1  What  is  the  last  term  equal  to  1  Give  the  rtdo  foi 
finding  the  last  term 


244  ELEMENTARY  ALGEBRA. 

EXAMPLES. 

1.  Find  the  5th  term  of  the  progression 

2  :  4  :  8  :   16  .  . 

in  which  the  first  term  is  2  and  the  common  ratio  2. 
5th  term=2x2*=2x  16=32     Ans. 

2.  Find  the  8th  term  of  the  progression 

2  :  6  :  18  :  54  ..." 

8th  term=2  x  37=2  x2187=4374     Ans. 

3.  Find  the  6th  term  of  the,  progression 

2  :  8  :  32  :  128  ... 
6th  term=2  x  45=2  x  1024=2048     Ans. 

4.  Find  the  7th  term  of  the  progression 

3  :  9  :  27  :  81   ... 

7th  term=3x36=3x  729=2187     Ans. 

5.  Find  the  6th  term  of  the  progression 

4  :  12  :  36  :  108  ... 
6th  term=4x35= 4x243  =  972     Ans. 

6.  A  person  agreed  to  pay  his  servant  1  cent  for  the  first 
day,  two  for  the  second,  and  four  for  the  third,  doubling 
every  day  for  ten  days  :  how  much  did  he  receive  on  the 
tenth  day?  Ans.  $5,12. 


GEOMETRICAL    PROGRESSION.  245 

7.  What  is  the  8th  term  of  the  progression 

9  :  36  :   144  :  576  ... 
8thterm=:9x47=:9xl6384=:147456     Ans. 
8    Find  the  12th  term  of  the  progression 

64  :  16  :  4  :  1   :  —  .  .  . 
4 

12th  termz=64( —  )    = = — = Ans. 


168.  We  will  now  proceed  to  determine  the  sum  of  n 
terms  of  the  progression 

a  :  b  :  c  :  d  :  e  :  f  :  .  .  .  :  i  '•  k  :  I', 

I  denoting  the  nth  term. 

We  have  the  equations  (Art.  167), 

b  —  aq,     c  =  bq,     d  =  cq,     e=dq,  .  .  .  k=iq,     l=kq\ 

and  by  adding  them  all  together,  member  to  member,  we 
deduce 

Sum  of  1st  members.  Sum  of  %nd  members. 

£+c+d+e+  .  .  .  +k+l=(a+b+c+d+  .  .  .  +»'+%; 

in  which  we  see  that  the  first  member  wants  the  first  term 
«,  and  the  polynomial  within  the  parenthesis  in  the  second 
member  wants  the  last  term  /.  Hence,  if  we  call  the  sum 
of  the  terms  S,  we  have 


whence 


S-a=(S— l)q=Sq— Iq,     or     S?— S  =  fy— a; 
Iq — a 


246  ELEMENTARY  ALGEBRA. 

Therefore,  to  obtain  the  sum  of  the  terms  of  a  geometrical 
progression,  we  have  the  following 


RULE. 

I.  Multiply  the  last  term  by  the  ratio. 

II.  Subtract  the  frst  term  from  the  product. 

III.  Divide  the  remainder  by  the  ratio  diminished  by  unity, 
and  the  quotient  will  be  the  sum  of  the  series. 

1  .  Find  the  sum  of  eight  terms  of  the  progression 
2  :  6  :   18  :  54  :   162  ...  2x37^4374. 


q-l  2 

2    Find  the  sum  of  the  progression 

2  :  4  :  8  :  16  :  32. 


q-\  1 

3.  Find  the  sum  of  ten  terms  of  the  progression 

2  :  6  :   18  :   54  :   162  ...  2x39  =  39366. 

Ans.  59048. 

4.  What  debt  may  be  discharged  in  a  year,  or  twelve 
months,  by  paying  $1  the  first  month,  $2  the  second  month, 


QUEST. — 168.  Give  the  rule  for  finding  the  sum  of  the  series.     What 
the  first  step  1     What  the  second  1     What  the  third  7 


GEOMETRICAL    PROGRESSION. 

$4  the  third  month,  and  so  on,  each  succeeding  payment 
being  double  the  last ;  and  what  will  be  the  last  payment  ? 

(  Debt,       .     .     $4095. 

!  Last  payment,  $2048. 

5.  A  gentleman  married  his  daughter  on  New  Year's  day, 
and  gave  her  husband  Is.  towards  her  portion,  and  was  to 
double  it  on  the  first  day  of  every  month  during  the  year : 
what  was  her  portion?  Ans.  £204:  15s. 

6.  A  man  bought   10  bushels  of  wheat  on  the  condition 
that  he  should  pay  1  cent  for  the  1st  bushel,  3  for  the  second, 
9  for  the  third,  and  so  on  to  the  last :  what  did  he  pay  for 
the  last  bushel  and  for  the  ten  bushels  ? 

*        (  Last  bushel,  $196,83. 
'*  \  Total  cost,     $295,24. 

7.  A  man  plants  4  bushels  of  barley,  which,  at  the  first 
harvest,  produced  32  bushels  ;  these  he  also  plants,  which, 
in  like  manner,  produce  8  fold  ;  he  again  plants  all  his  crop, 
and  again  gets  8  fold,  and  so  on  for  16  years :  what  is  his 
last  crop,  and  what  the  sum  of  the  series  ? 

Last,  140737488355328^. 
Sum,  16084284*3834660. 

169    Whep   the  progression  is   decreasing,   we  have 
and  l-^a ;  the  above  formula 

Iq— a 

?-i  ' 

for  the  sum  is  then  written  under  the  form 
g^  a— Iq 

in  order  that  the  two  terms  of  the  fraction  may  be  positive. 

QUEST. — 163.  What  is  the  formula  for  the  sum  of  the  series  of  a 
decreasing  progression  "? 


248  ELEMENTARY  ALGEBRA. 

1.  Find  the  sum  of  the  terms  of  the  progression 
32  :   16  :  8  :  4  :  2. 


31 


. 

l—q 

~2  ~2 

2.  Find  the  sum  of  the  first  twelve  terms  of  the  progression 


l-l  •  l  ' 

•eifM11      or          ] 

4 

1          1 
65536     4 

\4/    '              65536 

1 
65536               65535 

:  —  85  J  . 

4 

REMARK. —  17O.  We  perceive  that  the  principal  diffi- 
culty consists  in  obtaining  the  numerical  value  of  the  last 
term,  a  tedious  operation,  even  when  the  number  of  terms 
is  not  very  great. 

3.  Find  the  sum  of  6  terms  of  the  progression 

512  :  128  :  32  ... 

Ans.  682^ 

4.  Find  the  sum  of  seven  terms  of  the  progression 

2187  :  729  :  243  ... 

Ans.  3279. 

5.  Find  the  sum  of  six  terms  of  the  progression 

972  ;  324  :  108 

Ans.  1456. 

6.  Find  tho  sum  of  8  terms  of  the  progression 

147456  :  36864  :  9216  .  .  . 

Ans.  196605 


GEOMETRICAL    PROGRESSION.  249 

Of  Progressions  having  an  infinite  number  of  terms 

171.  Let  there  be  the  decreasing  progression 

a  :  b  :  c  :  d  :  e  :  f  :  .  .  . 

containing  an  indefinite  number  of  terms.     In  the  formula 

a-lq 

-- 


substitute  for  /  its  value  aqn~l  (Art.  167),  and  we  have 


which  represents  the  sum  of  n  terms  of  the  progression. 
This  may  be  put  under  the  form 


Now,  since  the  progression  is  decreasing,  q  is  a  proper 
fraction ;  and  qn  is  also  a  fraction,  which  diminishes  as  n 
increases.  Therefore,  the  greater  the  number  of  terms  we 
take,  the  more  will x  qn  diminish,  and  consequent- 
ly the  more  will  the  partial  sum  of  these  terms  approximate 

to  an  equality  with  the  first  part  of  S,  that  is,  to     . 

\-q 

Finally,  when  n  is  taken  greater  than  any  given  number,  or 
n = infinity,  then X^"     will  be  less  than  any  given 

number,  or  will  become  equal  to  0  ;  and  the  expression    


will  represent  the  true  value  of  the  sum  of  all  the  terms  of 

the  series.     Whence  we  may  conclude,  that  the  expression 

22* 


250  ELEMENTARY  ALGEBRA. 

for  the  sum  of  the  terms  of  a  decreasing  progression,  in  uihich 
the  number  of  terms  is  infinite,  is 


That  is,  equal  to  the  first  term  divided  by  1  minus  the  ratio. 

This  is,  properly  speaking,  the  limit  to  which  the  partial 
sums  approach,  by  taking  a  greater  number  of  terms  in  the 
progression.  The  difference  between  these  sums  and 

-  -     can  become  as  small  as  we  please,  and  will  only 
become  nothing  when  the  number  of  terms  taken  is  infinite 


EXAMPLES. 
1.  Find  the  sum  of 


I:T:T::§T  toinfinity- 

We  have  for  the  expression  of  the  sum  of  the  terms 


The  error  committed  by  taking  this  expression  for 
value  of  the  sum  of  the  n  first  terms,  is  expressed  by 

3/ 


First  take  n=5  ;  it  becomes 

3  /  1  x5          1  1 


2  \  3  /        2  .  3*        162 


QUEST. — 165.  When  the  progression  is  decreasing  and  the  number  o< 
terms  infinite,  what  is  the  value  of  the  sum  of  the  series  1 


GEOMETRICAL    PROGRESSION.  251 

When  n  =  6,  we  find 


2\3/        162       3       486 

q 

Whence  we  see  that  the  error  committed,  when     -~     is 

2 

taken  for  the  sum  of  a  certain  number  of  terms,  is  less  in 
proportion  as  this  number  is  greater. 
2.  Again  take  the  progression 

.1   •  J_      I.      JL      .1  •   & 
2    :    4    :    8    :  16  :  32  : 

We  have 


3.  What  is  the  sum  of  the  progression 

re-   W'   !»'   TdSoc 


'•-         '         '  'te  »»«"«>'• 


10 

172.  In  the  several  questions  of  geometrical  progres 
sion  there  are  five  numbers  to  be  considered  : 

1st.    The  first  term,       ........  a. 

2nd.  The  ratio,    ..........  q. 

3rd.    The  number  of  terms,     ......  n. 

4th.    The  last  term,       ........  I 

5th.    The  sum  of  the  terms,  ...  S. 


QUEST. — 166.  How  many  numbers  are  considered  in  geometrical  pro- 
gression?    What  are  they' 


252  ELEMENTARY  ALGEBRA. 

173.  We  shall  terminate  this  subject  by  the  question, 

To  find  a  mean  proportional  between  any  two  numbers, 
as  m  and  n. 

Denote  the  required  mean  by  x.     We  shall  then  have 
(Art.  156), 

xz= 

and  hence 


That  is,  Multiply  the  two  numbers  together,  and  extract  the 
square  root  of  the  product. 

1.  What  is  the  geometrical  mean  between  the  numbers 
2  and  8  ? 


Mean=  ^fSx2—  yT6—  4     Ans. 

2.  What  is  the  mean  between  4  and  16?  Ans.  8 

3.  What  is  the  mean  between  3  and  27  ?  Ans.  9. 

4.  What  is  the  mean  between  2  and  72?  Ans.   12. 

5.  What  is  the  mean  between  4  and  64  ?  Ans.   16. 


QUEST. — 167.  How  c.0  you  find  a  mean  proportional  between  -iwo 
numbers ' 


OF  LOGARITHMS,  253 


CHAPTER  VIII. 

Of  Logarithms. 

1  74.  The  nature  and  properties  of  the  logarithms  in  com 
tnon  use,  will  be  readily  understood,  by  considering  atten- 
tively the  different  powers  of  the  number  10.     They  are, 

10°  =  1 

10^  =  10 

102  =  100 

103  =  1000 

104  =  10000 

105  =  100000 
&c.  &c. 

It  is  plain  that  the  indices  or  exponents  0, 1,  2,  3,  4,  5,  &c. 
form  an  arithmetical  series  of  which  the  common  difference 
b  1 ;  and  that  the  numbers  1,  10,  100,  1000,  10000,  100000, 
&.c.  form  a  geometrical  series  of  which  the  common  ratio  is 
10.  The  number  10,  is  called  the  base  of  the  system  of  log- 
arithms ;  and  the  indices  0,  1,  2,  3,  4,  7,  &,c.,  are  the  loga- 

QrEsi. — 174.  What  relation  exists  between  the  exponents  1,  2,  3, 
&<•.?  How  .are  the  corresponding  numbers  10,  100,  10001  What  is 
the  common  difference  of  the  exponents'!  What  is  the  common  ratio  of 
the  corresponding  numbers  1  What  is  the  base  of  the  common  system 
of  logarithms]  What  are  the  indices?  Of  what  number  is  the  index 
1  the  logarithm  *  The  index  2  ?  The  index  3  ? 


254  ELEMENTARY  ALGEBRA. 

rithms  of  the  numbers  which  are  produced  by  raising  10  to 
the  powers  denoted  by  those  indices. 

1  75;  If  we  denote  the  base  of  the  system  by  a,  and  the 
logarithm  of  any  number  by  wi,  then  the  number  itself  will 
e  the  mth  power  of  a  :  that  is,  if  we  represent  the  corres- 
onding  number  by  M, 

a™=M 

Thus,  if  we  make  ?w=0,  M  will  be  equal  to  1  ;  if  m=>l, 
M  will  be  equal  to  10,  &c.     Hence, 

The  logarithm  of  a  number  is  the  exponent  of  the  power 
to  which  it  is  necessary  to  raise  the  base  of  the  system  in  order 
to  produce  the  number. 

1  7G.  Letting,  as  before,  a  denote  the  base  of  the  system 
of  logarithms,  m  any  exponent,  and  M  the  corresponding 
number  :  we  shall  then  have, 

am=M 
in  which  m  is  the  logarithm  of  M. 

If  we  take  a  second  exponent  w,  and  let  JV  denote  the  cor- 
responding number,  we  shall  have, 
a"=JV 
in  which  n  is  the  logarithm  of  JV. 

If  now,  we  multiply  the  first  of  these  equations  by  the 
second,  member  by  member,  we  have 


but  since  a  is  the  base  of  the  system,  m-\-n  is  the  logarithm 
hence, 


QUEST.  —  IfS.  If  we  denote  the  base  of  a  system  by  a,  and  the  expo- 
nent  by  m,  what  will  represent  the  corresponding  number  ?  What  is  the 
logarithm  of  a  number  1  176.  To  what  is  the  sum  of  the  logarithms  of 
any  two  numbers  equal  1  To  what  then,  will  the  addition  of  logarithms 
correspond  ? 


OF  LOUA.RITHM8.  .  255 

The  sum  of  the  logarithms  of  any  two  numbers  is  equal  to 
the  logarithm  of  their  product. 

Therefore,  the  addition  of  logarithms  corresponds  to  the 
multiplication  of  their  numbers. 

177.  If  we  divide  the  equations  by  each  other,  member 
by  member,  we  have, 

-W-J?; 

a"  JV' 

but  since  a  is  the  base  of  the  system,  m — n  is  the  logarithm 
of  — ;  hence, 

If  one  number  be  divided  by  another,  the  logarithm  of  the 
quotient  will  be  equal  to  the  logarithm  of  the  dividend  dimi- 
nished by  that  of  the  divisor. 

Therefore,  the  subtraction  of  logarithms  corresponds  to  the 
division  of  their  numbers. 

178.  Let  us  examine  further  the  equations 

10°  =1 

101=10 

103=100 

103=1000 

&c.       &c. 

It  is  plain  that  the  logarithm  of  1  is  0,  and  that  the  loga- 
rithms of  all  the  numbers  between  1  and  10,  are  greater  than 
0  and  less  than  1.  They  are  generally  expressed  by  decimal 
fractions :  thus, 

log  2=0.301030 

QUEST. — \*i*t  If  one  number  be  divided  by  another,  what  will  the 
logarithm  of  the  quotient  be  equal  tol  To  what  then  will  the  subtrac- 
tion of  logarithms  correspond  ]  1?§.  What  is  the  logarithm  of  1  ? 
Between  what  limits  are  the  logarithms  of  all  numbers  between  1  and  10  ] 
How  are  they  generally  expressed? 


256  ELEMENTARY  ALGEBRA. 

The  logarithms  of  all  the  numbers  greater  than  10  and  less 
than  100,  are  greater  than  1  and  less  than  2,  and  are  generally 
expressed  by  1  and  a  decimal  fraction  :  thus, 
log  50=1.698970. 

The  part  of  the  logarithm  which  stands  on  the  left  of  the 
decimal  point,  is  called  the  characteristic  of  the  logarithm. 
The  characteristic  is  always  one  less  than  the  places  of  integer 
figures  in  the  number  whose  logarithm  is  taken. 

Thus,  in  the  first  case,  for  numbers  between  1  and  10, 
there  is  but  one  place  of  figures,  and  the  characteristic  is  0. 
For  numbers  between  10  and  100,  there  are  two  places  of 
figures,  and  the  characteristic  is  1 ;  and  similarly  for  other 
numbers. 

Table  of  Logarithms. 

1 70.  A  table  of  logarithms  is  a  table  in  which  are  written 
the  logarithms  of  all  numbers  between  1  and  some  other 
given  number.  A  table  showing  the  logarithms  of  the  num- 
bers between  1  and  100  is  annexed.  The  numbers  are  written 
in  the  column  designated  by  the  letter  N,  and  the  logarithms 
in  the  columns  designated  by  Log. 

QUEST.— How  is  it  with  the  logarithms  of  numbers  between  10  and 
100"?  What  is  that  part  of  the  logarithm  called  which  stands  at  the  left 
of  the  characteristic?  What  is  the  value  of  the  characteristic?  IT'O- 
What  is  a  table  of  logarithms  1  Explain  the  manner  of  finding  the  loga- 
rithms of  numbers  between  1  and  100? 


OP  LOGARITHMS- 

TABLE. 


257 


N. 

Log. 

N. 

Log. 

N. 

Log. 

N. 

Log. 

1 
2 

3 
4 
5 

0.000000 
0.301030 
0.477121 
0.602060 
0.698970 

26 
27 

28 
29 
30 

1.414973 
1.431364 
1.447158 
1.462398 
1.477121 

51 
52 
53 
54 
55 

1.707570 
1.716003 
1.724276 
1.732394 
1.740363 

76 

77 
78 
79 
80 

1.880814 
1.886491 
1.892095 
1.897627 
1.903090 

6 

7 
8 
9 
10 

0.778151 
0.845098 
0.903090 
0.954243 

1.000000 

31 
32 
33 
34 
35 

1.491362 
1.505150 
1.518514 
1.531479 
1.544068 

56 
57 

58 
59 
60 

1.748188 
1.755875 
1.763428 
1.770852 
1.778151 

81 

82 
83 
84 

85 

1.908485 
1.913814 
1.919078 
1.924279 
1.929419 

11 
12 
13 
14 
15 

1.041393 
1.079181 
1.113943 
1.146128 
1.176091 

36 
37 
38 
39 
40 

1.556303 
1.568202 
1.579784 
1.591065 
1.602060 

61 
62 
63 
64 
65 

1.785330 
1.792392 
1.799341 
1.806180 
1.812913 

86 
87 
88 
89 
90 

1.934498 
1.939519 
1.944483 
1.949390 
1.954243 

16 
17 

18 
19 

20 

1.204120 
1.230449 
1.255273 
1.278754 
1,301030 

41 
42 
43 
44 
45 

1.612784 
1.623249 
1.633468 
1.643453 
1.653213 

66 
67 
68 
69 

70 

1.819544 
1.826075 
1.832509 
1.838849 
1.845098 

91 
92 
93 
94 
95 

1.959041 
1.963788 
1.968483 
1.973128 
1.977724 

21 
22 
23 
24 
25 

1.322219 
1.342423 
1.361728 
1.380211 
1.397940 

46 
47 
48 
49 
50 

1.662758 
1.672098 
1.681241 
1.690196 
1.698970 

71 
72 
73 
74 
75 

1.851258 
1.857333 
1.863323 
1.869232 
1.875061 

96 
97 
98 
99 
100 

1.982271 
1.986772 
1.991226 
1.995635 
2.000000 

EXAMPLES. 

1.  Let  it  be  required  to  multiply  8  by  9,  by  means  of  loga 
rithms.  We  have  seen,  Art.  176,  that  the  sum  of  the  loga- 
rithms is  equal  to  the  logarithm  of  the  product.  Therefore, 
find  the  logarithm  of  8  from  the  table,  which  is  0.903090 
and  then  the  logarithm  of  9,  which  is  0.954243;  and  their 
sum,  which  is  1.857333,  will  be  the  logarithm  of  the  product* 

23 


2.58  ELEMENTARY  AI  GEBRA 

In  searching  along  in  the  table,  we  find  that  72  stands  oppo 
site  this  logarithm :  hence,  72  is  the  product  of  8  by  9. 
2.  What  is  the  product  of  7  by  12  ? 

Logarithm  of  7  is,  .         .         .     0.845098 

Logarithm  of  12  is,     .         .         .         .     1.079181 


Logarithm  of  their  product,          .         .     1.924279 
and  the  number  corresponding  is  84. 
3.  What  is  the  product  of  9  by  11  ? 

Logarithm  of  9  is,       .         .         .  0.954243 

Logarithm  of  11  is,     .         .         .         .     1.041393 


Logarithm  of  their  product,          .         .     1.995636 
and  the  corresponding  number  is  99. 

4.  Let  it  be  required  to  divide  84  by  3.     We  have  seen  in 
Article  177,  that  the  subtraction  of  Logarithms  corresponds 
to  the  division  of  their  numbers.     Hence,  if  we  find  the  lo- 
garithm of  84,  and  then  subtract  from  it  the  logarithm  of   3, 
the  remainder  will  be  the  logarithm  of  the  quotient. 
The  logarithm  of  84  is,       .         .         .     1.924279 
The  logarithm  of  3  is,         .         .         .     0.477121 


Their  difference  is,  ...     1.447158 

and  the  number  corresponding  i§  28. 
5.  What  is  the  product  of  6  by  7  ? 

Logarithm  of  6  is,       .         .         .        .     0.778151 
Logarithm  of  7  is,       .         .         .         .     0.845098 


Their  sum  is, 1.623249 

and  the  corresponding  number  of  the  table,  42. 


SUPPLEMENT. 

EXAMPLES  IN  ADDITION     AND    SUBTRACTION 

1.  What  is  the  sum  of 

ax« 

2.  What  is  the  sum  of 


3.  What  is  the  SUIT,  of 

10a4  -f.  3a*+  Go4—  d4—  -M 

4.  What  is  the  sum  of 


5.  What  is  the  sum  of 

an&m—  9a"'-}~5ani-  +  6ara-(- 

6.  What  is  the  sum  of 


7.  What  is  the  sum  of 
5a*b  +  3a2b*c—  7ab—  Qa 


8.  What  is  the  sum  of 
3 


9.  What  is  the  sura  of 
9«W— 


10.  From          —  Qa"1^  —  13+2^^  —  46mco^ 
take  3bmcx*  — 


ELEMENTARY  ALGEBRA SUPPLEMENT. 

11.  From          5a4— 7a3b2— 3cd*+7d 
take  —l5</b2 + 3a4— 3a2— Vci/2 

12.  From          9am62+6a&m— d5+18a46n 
take  7a4b"+d5—8ub'»+9a™b2 

13.  From          1265<f—  16a868— 5am&n+6ac° 
take  6am6n— 6acb + 16a866  +  1265<Z6 

14.  From          8a?b3cs- 

take  Sa^-— 8aW—  12am^ 

15.  From          12am^n — 9oar5 — 4a6-|-6a2Z>2 — a 
take  3a— 6a262+  12am6n— 9a^-f  5* 

EXAMPLES  IN  MULTIPLICATION. 

1.  What  is  the  product  of 

2.  What  is  the  product  of 

2a*x7a9x— 3a* 

3.  What  is  the  product  of 


4.  What  is  the  product  of 

_-aP-i  x  —  SaP-2  X/X  Sa^+'c 

5.  What  is  the  product  of 

5a364xl0a265cx—  3a7 

6.  What  is  the  product  of 


7.  What  is  the  product  of 

ambpcq  X  an6rcq  X  an'6  X 

8.  What  is  the  product  of 

(a2— 

9.  What  is  the  product  of 

(2a3//— 


EXAMPLES  IN  MULTIPLICATION.  261 

10.  What  is  the  product  of 


11.  What  is  the  product  of 

(aW—cb5d3f+  3cm)  X— 

12.  What  is,  the  product  of 


13.  What  is  the  product  of 

(3F—  5M+2P)  x(#—  7kl) 

14.  What  is  the  product  of 


15.  What  is  the  product  of 

(4a2—  16aa?+  Sz2)  x  (So3—  2«2z)« 

16.  What  is  the  product  of 


17.  What  is  the  product  of 


18.  What  is  the  product  of 

(2aV—  3  jy  )  x  (2<*V  -h  35y  ) 

19.  What  is  the  product  of 

+6at2—2b3)  X(3a4—  4a3J-f 

20.  What  is  the  product  of 


EXAMPLES  IN  DIVISION. 

1.  Divide   am   by   an. 

2.  Divide   am   by   a2". 

3.  Divide   8a16   by   2O4. 

4.  Divide   ca18   by   rfa4. 

23* 


262  ELEMENTARY  ALGEBRA  -  SUPPLEMENT. 


5.  Divide  6(a+Z>)9   by 

6.  Divide  (a+x)2x(a+y)3   by    (a+x)x(a+y)2. 

7.  Divide  6a3£2—  15a2/+27a4fo,   by    3a2. 

8.  Divide  be3  —  csx  by    I—  x. 

9.  Divide  a3+a2&—  atf—  -W   by   a—  £. 

10.  Divide  3a6+16a45—33a3i2-hl4^3   by    a2-f  7a£. 

11.  Divide  a7—  6a6£3+14asi6—  12a4Z>9   by   a3—  2a2£3. 

12.  Divide  a4—  2a262+£4   by    a2—  J2. 

13.  Divide  —  a*b*+15antf—  48a14Z»5—  20ani7 
by    lOa9*2—  a6*. 

14.  Divide   a8—  16*8   by   a2—  2«2. 

15.  Divide  2a4—  13a3J+31a2^—  38ai3+24J4 
by  202—  3ai+4^ 

16.  Divide  4c4—  9&V+6£3c—  fl*   by   2c2—  3Jc  f  P. 

17.  Divide  —  l+aV   by   —  1  +  an. 

18.  Divide  ae+2aV+z6   by   a2  —  02  +  a*. 

19.  Divide  i—  6z2+27^   by   i-f-22  +  322. 

20.  Divide  a6—  16aV+64a?6   by 

21.  Divide   a3d3—  3a2cd3+3ac'^3 
by    a2^_  2acd*+c2d?+aczd. 

EXAMPLES  IN  REDUCTION  OF  FRACTIONS. 

1.  Reduce  to  its  simplest  terms  the  fraction 

ISacf—  Qbdcf—  2ad 
3adf 

2.  Reduce  to  its  simplest  terms  the  fraction 


—  2a 


EXAMPLES  IN  REDUCTION  OF  FRACTIONS.  263 

3.  Reduce  to  its  simplest  tetms  the  fraction 
12ac 


4.  Reduce  to  its  simplest  terms  the  fraction 
ab  —  ac 
b—c 


5.  Reduce    a — 6-f-— —    to  the  form  of  a  fraction. 
x — a 


6.  Reduce    x —  L.    to  the  form  of  a  fraction. 

ax — c 


7.  Reduce    a-f  ft-f-  to  the  form  of  a  fraction. 


8.  Reduce    x  —  ab  —  -  --    to  the  form  of  a  fraction. 
r,  —  x 


9.  Reduce    a  —  l_n       to  the  form  of  a  fraction. 


10.  Reduce    6«/2ar-f  9a/—  -+ax  to  the  form  of  a  fraction. 


11.  Reduce    5acx  —  y  —  _  --   to  the  form  of  a  fraction. 

fat 

12.  Reduce  to  an  entire  or  mixed  quantity  the  fraction 

fee 


2a2 

13.  Reduce  to  an  entire  or  mixed  quantity  the  fraction 


*2(54  ELEMENTARY  ALGEBRA  -  SUPPLEMENT. 

14.  Reduce  to  an  entire  or  mixed  quantity  the  fraction 

a3-—  2a*x+ab+ax*-bx 
a  —  x 

15.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

a  —  x     b  —  c     &ax  —  e 

—  .    ,    —  _  —  ,    ________  . 

a+x      f        a  —  x 

16.  Reduce  the  following  fractions  to  a  common  denomi- 
nator: viz.        • 

_JL_,   ?H?and-_£_ 
3ax—  b       3Z>  a—x 

17.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 


7a  —  c  c  y 

18.  Reduce  the  following  fractions  to  a  common  denomi- 
nator :  viz. 

*"+*    and   8ac-/ 
Sac  —  -f  4a# 

19.  Reduce  the  following  fractions  to  a  common  denomi- 
nator: viz. 


a  —  x      a+x  a 

20.  Reduce  the  following  fractions  to  a  common  denomi- 
nator: viz. 

and   ±1? 


x  —  a  c  x-\-a 


EXAMPLES  IN  ADDITION  AND  SUBTRACTION.  265 


ADDITION,    SUBTRACTION,  MULTIPLICATION    AND    DIVISION 
OF    FRACTIONS. 

1.  What  is  the  sum  of    —2—,   __£_  and  c 
x — a    a — x 


2.  What  is  the  sum  of    *""*,    ^^  and  y 
b        a — c 


3.  What  is  the  sum  of    ??,   ±=2?,   - 

b         d        3c— / 

4.  What  is  the  sum  of     ax—f,    ac—g,     day 

c+a 


5.  What  is  the  sum  of    3a-f  ,   2a—  —    . 

x  —  a  a+b 


6.  From   8a+  -    take 

b  a  —  i 


7.  From   ^~^    take    3— 

b  c 


8.  From  take 


ay 


9.  From  ay-  "*<     take    3ay+ 
oa? 


10.  From  take   7aJ»  - 

ga?  8a—  a? 


11.  Multiply    7a+          by 
6 


12.  Multiply    _L     by    3ar/-~ 
a;  —  a 


266  ELEMENTARY  ALGEBRA SUPPLEMENT. 

13   Multiply     9a~a?     by  2a+ 

14.  Multiply   3ax+     5ay-~x    by  5+  -fL 

— y  & 

15.  Multiply   6a+    ^"""8a     by   6a~"* 

— b  a — lr 

16.  Divide   Sac— 2adc— /+  ±   by  2a 

17.  Divide     ?-h^—  Sac 4-7  by    ?f 

a      2c  d 

18.  Divide  3a2-  M  -  ?lf£  -  _^+  ?^  -  ^ 

24282 

by   3a— 5£+  ?£ 


19.  Divide        -  --- 

12          9  8 


20.  Divide  -         +          -          +          +26fJ 
"334 


by  ~ 


EXAMPLES  IN  EQUATIONS  OP  THE   FIRST  DEGREE. 


1.  Given   •{  }.  to  find  a:  and  y. 

£Z?'4-.ti  =41 


EXAMPLES  IN  EQUATIONS  OF  THE  FIRST  DEGREE.      267 


Given   <|  to  find  x  and  y. 


I  to 


3   Given  ax+JL-  -f  j  (#— /)  =  g  to  find  a?, 
"c — a 


4.  Given  """" *  +<"-  -^u~" *  „  «J,  to  find  ar. 
4  5  c 


5.  Given    J  I  to  find  x  and 


6.  Given  J  «^  i  to  find  x  and  y. 

/     /-»  r*  t>— ~7/  * 


3 


7.  Given  <  .  I  to  find  ar  and  y. 


8.  Given  ±Zf_  —  -f  x  «.  ^,  to  find  x. 

9.  Given  3a~6a?  ~  2a~3x  +  x  -  d  »/,  to  find  «. 


10.  Given 


+^  Z  ^  }  to  find  x  and  y. 


11.  Given  ±ll  +  ±=L  +  4(^—3)  =  68,  to  find  x. 

«J  <6 


ELEMENTARY  ALGEBRA  -  SUPPLEMENT. 


12.  Given  ~3~  to  find  «?,  y,  and  2. 


(  or  +  2y  +3Z  =  14  } 
13.  Given   J  *—  y  +  2  =  2  \  to  find  ar,  y,  and  z. 


x—  y  +  g+g  _  4 
14.  Given   ^4  5  I  to  find  *,  y  and  2. 


17.  Given   J  /^  =  ^  j  to  find  x  and  y. 

18.  Given    j«+        -  |    to  find  *  and 


C  _a_        _i       1 

19.  Given  <  J+y      3a+:r    >  to  find  x  and  y. 

(  a.r+2  ly  =  d    ) 

C  lex  =  cy  —  2J  J 

20.  Given   j  ^  +  a(c3-y)  =  2^          ^    [  to  find  a;  and  y. 


21.  Given  %  / 

2bf2  }  to  find  x  and  y. 

I  y — x= — — sL 

V—  f*  r 

J  j 


EXAMPLES  IN  EQUATION?.  OF  THE  FIRST  DEGREE.      269 


'C  x  +  y  +  z  =  29H 
22.  Given    <  x  +  y  —  z  =  18j  >  to  find  a:,  y  and  z. 

i  i  1Ox  m 

(a?  —  y  +  z  =  13|  ) 


23.  Given 


(3*  + 
iven   <  7x  + 
(2y  + 


%  =  161  ) 

2z  =  209  >  to  find  a;,  y  and  z. 

2  =  89 


24.  Given   < 


1,1 

-  +  - 

x       z 


to  find  a?,  y  and 


25.  Given   ^  dx  +  ey  =  f  ^  to  find  a?,  y  and 

gy  +  ^  = 


EXAMPLES  IN  EQUATIONS  OF  THE  SECOND  DEGREE. 

1.  Given  ar2— 5fz=18  to  find  x. 

2.  Given  Sz2— 2#=65  to  find  a?. 

3.  Given  622^=15^+6384  to  find  x. 

4.  Given  ll$x—3%3?= — 41J,  to  find  x. 

5.  Given  9^—901*+ 195=0,  to  find  x. 

6.  Given  20748— 1616a?+21^=0,  to  find  x. 

7.  Given  9^—90^+195=0,  to  find  x. 

18078* 


8.  Given  .+  +  4728=0,  to  find  x. 


65 


24 


270  ELEMENTARY  ALGEBRA-SUPPLEMENT. 

9.  Given  x2  —  Sx=  14  to  find  x. 

10.  Given  3z2-f  x=7  to  find  x. 

11.  Given  1182—2^=20  to  find  ar. 

12.  Given  6:r—  30=3^  to  find  x. 

13.  Given  8^—7^4-34=0 

14.  Given  4i^  —  9#=5#2  —  255f  —  Sx  to  find  x. 


15.  Given     80*+—  +glgir 

16.  Given         -..     l  to  find 


17.  Given  --  to  find  ,. 

10^—81       5a?—  8     5 


18.  Given     A0"ra>     ^^T^     w    _  to  find  x. 

6(3— x)      19— 7x    4(3— #) 

19.  Given     #2—  7a?-f3J=0 

20.  Given    4^— 9a?=5a^— 255|— 8a?  to  find  x. 

21.  Given     — ^-= — - —  to  find  x. 

3X 5 


22.  Given    -12-  +??=13  to  find  a?. 

x — 5     x' 

23.  Given     — — 6=?2  to  find  *. 

x+2          3x 

24.  Given     iL  =  J^-— 5  to  find  ». 


271 


PROMISCUOUS   QUESTIONS. 

EQUATIONS    OF   THE    FIRST    DEGREE. 

1 .  A  person  expends  30  cents  for  apples  and  pears,  giving 
one  cent  for  four  apples,  and  one  cent  for  five  pears  :  he  then 
sold,  at  the  prices  he  gave,  half  his  apples  and  one-third  his 
pears,  for  13  cents.     How  many  did  he  buy  of  each  ? 

2.  A  tailor  cut  19  yards  from  each  of  three  equal  pieces 
of  cloth,  and  17  yards  from  another  of  the  same  length,  and 
found  that  the  four  remnants  were  altogether  equal  to  142 
yards.     How  many  yards  in  each  piece  ? 

3.  A  fortress  is  garrisoned  by  2600  men,  consisting  of 
infantry,  artillery,  and  cavalry.     Now,  there  are  nine  times 
as  many  infantry,  and  three  times  as  many  artillery  soldiers, 
as  there  are  cavalry.     How  many  are  there  of  each  corps  ? 

4.  All  the  journey  ings  of  an  individual  amounted  to  2970 
miles.     Of  these  he  travelled  3J  times  more  by  water  than 
on  horseback,  and  2J  times  more  on  foot  than  by  water. 
How  many  miles  did  he  travel  in  each  way  ? 

5.  A  sum  of  money  was  divided  between  two  persons,  A 
and  B.     A's  share  was  to  exceed  B's  in  the  proportion  of  5  to 
3,  and  to  exceed  five-ninths  of  the  entire  sum  by  50.     What 
was  the  share  of  each  ? 

6.  There  are  52  pieces  of  money  in  each  of  two  bags,  out 
of  which  A  and  B  help  themselves.     A  takes  twice  as  much 


272  ELEMENTARY  ALGKBRA SUPPLEMENT. 

as  B  left,  and  B  takes  seven  times  as  much  as  A  left.     How 

much  did  each  take  ? 

x     t 

7.  Two  persons,  A  and  B,  agree  to  purchase  a  house  to 
gether,  worth  $1200.     Says  A  to  B,  give  me  two-thirds  of 
your  money  and  I  can  purchase  it  alone;  but,  says  B  to  A, 
if  you  give  me  three-fourths  of  your  money  I  shall  be  able  to 
purchase  it  alone.     How  much  had  each  ? 

8.  A  father  directs  that  $1170  shall  be  divided  among  his 
three  sons,  in  proportion  to  their  ages.     The  oldest  is  twice 
as  old  as  the  youngest,  and  the  second  is  one-third  older  than 
the  youngest.     How  much  was  each  to  receive  ? 

9.  Three  regiments  are  to  furnish  594  men,  and  each  to 
furnish  in  proportion  to  its  strength.     Now,  the  strength  of 
the  first  is  to  the  second  as  3  to  5 ;  and  that  of  the  second  to 
the  third  as  8  to  7  ?     How  many  must  each  furnish  ? 

10.  A  grocer  finds  that  if  he  mixes  sherry  and  brandy  in 
the  proportion  of  2  to  1,  the  mixture  will  be  worth  78s.  per 
dozen ;  but  if  he  mixes  them  in  the  proportion  of  7  to  2,  he 
can  get  79s.  a  dozen.     What  is  the  pri'ce   of  each  liquor  per 
dozen  ? 

11.  A  person  bought  7  books,  the  prices  of  which  were  in 
arithmetical  progression,  (in  shillings.)     The  price  of  the  one 
next  above  the  cheapest,  was  8  shillings,  and  the  price  of  the 
dearest,  23  shillings.     What  was  the  price  of  each  book  ? 

12.  A  number  consists  of  three  digits,  which  are  in  arith- 
metical proportion.     If  the  number  be  divided  by  the  sum  of 
the  digits,  the  quotient  will  be  26  -,  but,  if  198  be  added  to  it, 
the  digits  will  be  inverted. 

13.  A  person  has  three  horses,  and  a  saddle  which  is  worth 
$220.     If  the  saddle  be  put  on  the  back  of  the  first  horse,  it 


EQUATIONS  OF  THE  EIRST  DEGREE.  273 

will  make  his  value  equal  to  that  of  the  second  and  third ;  if 
it  be  put  on  the  back  of  the  second,  it  will  make  his  value 
double  that  of  the  first  and  third ;  if  it  be  put  on  the  back  of 
the  third,  it  will  make  his  value  triple  that  of  the  first  and 
second.  What  is  the  value  of  each  horse  ? 

14.  The  crew  of  a  ship  consisted  of  her  complement  of 
sailors,  and  a  number  of  soldiers.     There  are  22  sailors  to 
every  three  guns,  and  10  over ;  also,  the  whole  number  of 
hands  is  five  times  the  number  of  soldiers  and  guns  together. 
But  after  an  engagement,  in  which  the  slain  were  one-fourth 
of  the  survivors,  there  wanted  5  men  to  make   13  men  to , 
every  two  guns.     Required,  the  number  of  guns,  soldiers, 
and  sailors. 

15.  Three  persons  have  $96,  which  they  wish  to  divide 
equally  between  them.     In  order  to  do  this.  A,  who  has  the 
most,  gives  to  B  and  C  as  much  as  they  have  already :  then 
B  divides  with  A  and  C  in  the  same  manner,  that  is,  by  giving 
to  each  as  much  as  he  had  after  A  had  divided  with  them:  C 
then  makes  a  division  with  A  and  B,  when  it  is  found   that 
they  all  have  equal  sums.     How  much  had  each  at  first  ? 

16.  To  divide  the  number  a  into  three  such  parts,  that  the 
first  shall  he  to  the  second  as  m  to  n,  and  the  second  to  the 
third  as  p  to  q. 

17.  Five  heirs,  A,  B,  C,  D,  and  E,  are  to  divide  an  in- 
heritance of  $5600.     B  is  to  receive  twice  as  much  as  A,  and 
$,200  more;  C  three  times  as  much  as  A,  less  $400;  D  the 
half  of  what  B  and  C  receive  together,  and  150  more ;  and  E 
the  fourth  part  of  what  the  four  others  get,  plus  $475.     How 
much  did  each  receive? 

18.  A   person  has  four  casks,  the  second  of  which   being 

24* 


274 


ELEMENTARY    ALGEBRA SUPPLEMENT. 


filled  from  the  first,  leaves  the  first  four-sevenths  full.  The 
third  being  filled  from  the  second,  leaves  it  one-fomrth  full, 
and  when  the  third  is  emptied  into  the  fourth,  it  is  found  to 
fill  only  nine-sixteenths  of  it.  But  the  first  will  fill  the  third 
arid  fourth,  and  leave  15  quarts  remaining.  How  many 
quarts  does  each  hold  ? 

19.  A  courier  who  had  started  from  a  place  10  days,  was 
pursued  by  a  second  courier.     The  first  travels  4  miles  a 
day,  the  other  9.     How  many  days  before  the  second  will 
overtake  the  first? 

20.  If  the  first  courier  had  left  n  days  before  the  other, 
and  made  a  miles  a  day,  and  the  second  courier  had  travelled 
b  miles,  how  many  days  before  the  second  would  have  over- 
taken the  first  ? 

21.  A  courier  goes  31|  miles   every  five   hours,  and  is 
followed  by  another  after  he   had  been    gone  eight  hours. 
The   second   travels   22J   miles    every    three   hours.     How 
many  hours  before  he  will  overtake  the  first  ? 

22.  Two  places  are  eighty  miles  apart,  and  a  person  leaves 
one  of  them  and  travels  towards  the  other,  at  the  rate  of  3 J 
miles  per  hour.     Eight  hours  after,  a  person  departs  from  the 
second  place,  and  travels  at  the  rate  of  5^  miles  per  hour. 
How  long  before  they  will  meet  each  other  ? 

23.  Three  masons,  A,  B,  and  C,  are  to  build  a  wall.     A 
and  B  together  can  do  it  in  12  days ;  B  and  C  in  20  days ; 
and  A  and  C  in  15  days.     In  what  time  can  each  do  it  alone, 
and  in  what  time  can  they  all  do  it  if  they  work  together? 

24.  A  laborer  can  do  a  certain  work  expressed  by  a,  in  a 
time  expressed  by  b ;  a  second  laborer,  the  work  c  in  a  time 
'/ ;  a  third,  the  work  e  in  a  time  f.     It  is  required  to  find  the 


EQUATIONS^  OF  THE  FIRST  DEGREE.  275 

time  it  would  take  the  three  laborers,  working  together,  to 
perform  the  work  g. 

25.  Required  to  find  three  numbers    with  the   following 
conditions.     If  6  be  added  to  the  1st  and  2d,  the  sums  are  to 
one  another  as  2  to  3.     If  5  be  added  to  the  1st  and  3d,  the 
sums  are  as  7  to  11;  but,  if  36  be  subtracted  from  the  2d 
and  3d,  the  remainders  will  be  as  6  to  7. 

26.  The  sum  of  $500  was  put   out  at   interest,  in    two 
separate  sums,  the  smaller  sum  at  two  per  cent,  more  than 
the  other.     The  interest  of  the  larger  sum  was  afterwards 
increased,  and  that  of  the  smaller  diminished,  by  one  per 
cent.     By  this,  the  interest  of  the  whole  was  augmented  one- 
fourth.     But  if  the  interest  of  the  greater  sum  had  been  so 
increased,  without  any  diminution  of  the  less,  the  interest  of 
the  whole  would  have  been  increased  one-third.     What  were 
the  sums,  and  what  the  rate  per  cent.  ? 

27.  The  ingredients   of  a  loaf  of  bread  weighing  15lbs. 
are  rice,  flour,  and  water.     The  weight  of  the  rice,  augmented 
by  5lbs.,  is  two-thirds  the  weight   of  the   flour;    and   the 
weight  of  the  water  is  one-fifth  the  weight  of  the  flour  and 
rice  together.     Required,  the  weight  of  each. 

28.  Several  detachments  of  artillery  divided  a  certain  num- 
ber of  cannon  balls.     The  first  took  72  and   £   of  the  re- 
mainder; the  next  144  and  -£  of  the  remainder;    the  third 
216  and  |  of  the  remainder;  the  fourth  288  and  i  of  what 
was  left ;  and  so  on,  until  nothing  remained ;  when  it  was 
found  that  the  balls   were  equally  divided.      Required,  the 
number  of  balls  ind  the  number  of  detachments 

29.  A  banker  has  two  kinds  of  money;  it  takes  a  pieces 


276  ELEMENTARY  ALGEBRA SUPPLEMENT. 

of  the  first  to  make  a  crown,  and  b  of  the  second  to  make 
the  same  sum.  He  is  offered  a  crown  for  c  pieces.  How 
many  of  each  kind  must  he  give  ? 

30.  Find  what  each  of  three  persons,  A,  B,  and  C  is  worth, 
knowing,  1st,  that  what  A  is  worth,  added  to  Z  times  what 
B  and  C  are  worth,  is  equal  to  p  ;  2d,  that  what  B  is  worth, 
added  to  m  times  what  A  and  C  are  worth,  is  equal  to  q;  3d, 
that  what  C  is  worth,  added  to  n  times  what  A  and  B  are 
worth,  is  equal  to  r. 

31.  Find  the  values  of  the  estates  of  six  persons,  A,  B,  C, 

D,  E,  and  F,  from  the  following  conditions.     1st.  The  sum 
of  the  estates  of  A  and  B  is  equal  to  a ;  that  of  C  and  D  to 
b ;  and  that  of  E  and  F  to  c.     2d.  The  estate  of  A  is  worth 
m  times  that  of  C ;  the  estate  of  D  is  worth  n  times  that  of 

E,  and  the  estate  of  F  is  worth  p  times  that  of  B. 


277 


PROMISCUOUS   QUESTIONS. 


INVOLVING    EQUATIONS    OP    THE    SECOND    DEGREE. 

1.  FIND  three  numbers,  such,  that  the  difference  between 
the  third  and  second  shall  exceed  the  difference  between  the 
second  and  first  by  6 :  that  the  sum  of  the  numbers  shall  be 
33,  and  the  sum  of  their  squares  467. 

2.  It  is  required  to  find  three  numbers  in  geometrical  pro- 
gression, such  that  their  sum  shall  be  14,  and  the  sum  of 
their  squares  84. 

3.  What  two  numbers  are  those,  whose  sum  multiplied  by 
the  greater,  gives  144,  and  whose  difference  multiplied  by  the 
less,  gives  14  ? 

4.  What  two  numbers  are  those,  which  are  to  each  other 
as  m  to  ?i,  and  the  sum  of  whose  squares  is  b  ? 

5.  What  two  numbers  are  those,  which  are  to  each  other 
as  m  to  w,  and  the  difference  of  whose  squares  is  b  ? 

6.  A  certain  capital  is  out  at  4  per  cent  interest.     If  we 
multiply  the  number  of  dollars  in  the  capital  by  the  number 
of  dollars  in  the  interest,  for  five  months,  we  obtain  $117041|' 
What  is  the  capital? 

7.  A  person  has  three  kinds  of  goods,  which  together  cost 
$230/T.     One  pound  of  each  article  costs  as  many  times  ^ 
of  a  dollar  as  there  are  pounds  of  that  article.     Now,  he  has 


278  ELEMENTARY  ALGEBRA SUPPLEMENT. 

one-third  more  of  the  second  kind  than  of  the  first,  and  3j 
times  more  of  the  third  than  of  the  t  jcond.  flow  many 
pounds  had  he  of  each  ? 

8.  Required  to  find  three  numbers,  such,  that  the  producl 
of  the  first  and  second  shall  be  equal  to  a ;  the  product  of  the 
first  and  third  equal  to  b ;  and  the  sum  of  the  squares  of  the 
second  and  third  equal  to  c. 

9.  It  is  required  to  find  three  numbers,  whose  sum  shall  be 
38,  the  sum  of  their  squares  634,  and  the  difference  between 
the  second  and  first  greater  by  7  than  the  difference  between 
the  third  and  second. 

10.  Find  three  numbers  in  geometrical  progression,  whose 
sum  shall  be  52,  and  the  sum  of  the  extremes  to  the  mean, 
as  10  to  3. 

11.  The  sum  of  three  numbers  in  geometrical  progression 
is  13,  and  the  product  of  the  mean  by  the  sum  of  the  ex- 
tremes is  30.     What  are  the  numbers  ? 

12.  It  is  required  to  find  three  numbers,  such,  that  the 
product  of  the  first  and  second,  added  to  the  sum  of  their 
squares,  shall  be  37 ;  and  the  product  of  the  first  and  third, 
added  to  the  sum  of  their  squares,  shall  be  49 ;  and  the  pro- 
duct of  the  second  and  third,  added  to  the  sum    of  their 
squares,  shall  be  61. 

14.  Find  two  numbers,  such,  that  their  difference,  added 
to  the  difference  of  their  squares,  shall  be  -equal  to  150,  and 
whose  sum,  added  to  the  sum  of  their  squares,  shall  be  equal 
to  330. 

15.  It  is  required  to  find  a  number  consisting   of  three 
digits,  such,  that  the  sum  of  the  squares  of  the  digits  shall  be 


EQUATIONS    OF    THE    SECOND    DEGREE.  279 

04  ;  the  square  of  the  middle  digit  to  exceed  twice  the 
product  of  the  other  two  by  4  ;  and  if  594  be  subtracted  from 
the  number,  the  three  digits  become  inverted. 

16.  The  sum  of  two  numbers  and  the  sum  of  their  squares 
being  given,  to  find  the  numbers. 

17.  The  sum,  and  the  sum  of  the  cubes,  of  two  numbers 
being  given,  to  find  the  numbers. 

18.  To  find  three  numbers  irv arithmetical  progression  such, 
that  their  sum  shall  be  equal  to  18,  and  the  product  of  the 
two  extremes  added  to  25  shall  be  equal  to  the  square  of  the 
mean. 

19.  Having  given  the  sum,  and  the  sum  of  the  fourth  pow- 
ers of  two  numbers  ;  to  find  the  numbers. 

20.  To   find  three    numbers  in  arithmetical   progression, 
such,  that  the  sum  of  their  squares  shall  be  equal  to  1232, 
and  the  square  of  the  mean  greater  than  the  product  of  the 
two  extremes,  by  16. 

21.  To  find  two"  numbers  whose  sum  is  80,  and  such,  that 
if  they  be  divided  alternately  by  each  other,  the  sum  of  the 
quotients  shall  be  3i. 

22.  To  find  two  numbers  whose   difference  shall  be  10, 
and  if  600  be  divided  by  each  of  them,  the  difference  of  the 
quotients  shall  also  be  10. 


02512. 

.'850 


